Two-variable linear equations and their graphs

WEBVTT
Kind: captions
Language: en

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- [Voiceover] What I'd like to
introduce you to in this video

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is the idea of a Linear Equation.

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And just to start ourselves out,

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let's look at some examples
of linear equations.

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So, for example the equation

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y is equal to two x minus three,

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this is a linear equation.

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Now why do we call it a linear equation?

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Well if you were to take
the set of all of the

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xy pairs that satisfy this equation

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and if you were to graph
them on the coordinate plane,

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you would actually get a line.

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That's why it's called a linear equation.

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And let's actually feel
good about that statement.

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Let's see, let's plot some of the xy pairs

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that satisfy this equation
and then feel good

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that it does indeed generate a line.

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So, I'm just gonna pick some x values and

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make it easy to calculate
the corresponding y values.

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So, if x is equal to zero

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y is gonna be two times zero minus three

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which is negative three.

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And on our coordinate plane here that's--

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we're gonna move zero in the x direction,

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zero in the horizontal direction

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and we're gonna go down three
in the vertical direction,

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in the y direction.

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So, that's that point there.

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If x is equal to one, what is y equal to?

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Well two times one is two,
minus three is negative one.

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So we move positive one in the x direction

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and negative one, or down
one, in the y direction.

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Now let's see, if x is
equal to two what is y?

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Two times two is four, minus three is one.

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When x is equal to two y is equal to one.

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And hopefully you're seeing now

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that if I were to keep going,

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and I encourage you though if you want

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pause the video and try x equals three

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or x equals negative one and keep going.

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You will see that this is
going to generate a line.

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And in fact, let me connect these dots

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and you will see the line
that I'm talking about.

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So, let me see if I can draw,

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I'm gonna use the line tool here.

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Try to connect the dots
as neatly as I can.

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There you go.

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This line that I have just drawn,

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this is the graph,

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this is the graph of y is
equal to two x minus three.

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So if you were to graph all
of the xy pairs that satisfy

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this equation you are gonna get this line.

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And you might be saying,
"Hey wait wait, hold on Sal,

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you just tried some particular points,

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why don't I just get a bunch of points,

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how do I actually get a line?"

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Well, I just tried,

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over here I just tried
integer values of x.

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But you can try any value in
between here, all of these,

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it's actually a pretty unique concept.

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Any value of x that you input into this,

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you find the corresponding value for y,

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it will sit on this line.

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So for example,

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for example,

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if we were to take x is equal to,

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actually let's say x is
equal to negative point five.

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So if x is equal to negative point five

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if we look at the line

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when x is equal to negative point five

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it looks like

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it looks like y is equal to negative four.

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And that looks like that sits on the line.

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Well let's verify that.

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If x is equal to negative,

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I'll write that as negative one half,

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then what is y equal to?

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Let's see, two times negative one half,

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I'll just write it out,

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two times negative--

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two times negative one half minus three.

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Well this says two times
negative one half is

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negative one minus three
is indeed negative four

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it is indeed negative four.

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So you can literally take any, any--

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for any x value that you put here

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and the corresponding y value

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it is going to sit on the line.

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This point right over
here represents a solution

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to this linear equation.

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Let me do this in a color you can see.

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So this point represents a
solution to a linear equation.

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This point represents a
solution to a linear equation.

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This point is not a solution
to a linear equation.

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So if ex is equal to five

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then y is not gonna be equal to three.

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If x is gonna be equal to
five you go to the line to see

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what the solution to
the linear equation is.

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If x is five this shows us
that y is going to be seven.

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And it's indeed--

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that's indeed the case.

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Two times five is ten,
minus three is seven.

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The point--

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the point five comma seven is on,

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or it satisfies this linear equation.

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So if you take all of the
xy pairs that satisfy it,

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you get a line.

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That is why it is called
a linear equation.

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Now, this isn't the only
way that we could write

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a linear equation.

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You could write a linear equation like--

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let me do this in a new color.

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You could write a linear
equation like this:

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Four x minus three y is equal to twelve.

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This also is a linear equation.

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And we can see that if we
were to graph the xy pairs

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that satisfy this we would
once again get a line.

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X and y.

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If x is equal to zero,

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then this goes away and

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you have negative three
y is equal to twelve.

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Let's see, if negative
three y equals twelve

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then y would be equal to negative four.

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Nega-- zero comma negative four.

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You can verify that.

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Four times zero minus
three times negative four

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well that's gonna be
equal to positive twelve.

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And let's see, if y were to equal zero,

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if y were to equal zero

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then this is gonna be four
times x is equal to twelve,

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well then x is equal to three.

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And so you have the point
zero comma negative four,

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zero comma negative four on this line,

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and you have the point three
comma zero on this line.

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Three comma zero.

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Did I do that right?

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Yep.

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So zero comma negative four
and then three comma zero.

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These are going to be on this line.

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Three comma zero is also on this line.

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So this is, this line is
going to look something like--

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something like, I'll
just try to hand draw it.

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Something like that.

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So once again, all of the xy--

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all of the xy pairs that satisfy this,

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if you were to plot them
out it forms a line.

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Now what are some examples,
maybe you're saying

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"Wait, wait, wait, isn't any
equation a linear equation?"

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And the simple answer is
"No, not any equation is

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a linear equation."

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I'll give you some examples
of non-linear equations.

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So a non--

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non-linear,

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whoops let me write a
little bit neater than that.

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Non-linear equations.

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Well, those could include something like

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y is equal to x-squared.

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If you graph this

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you will see that this
is going to be a curve.

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it could be something like x
times y is equal to twelve.

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This is also not going to be a line.

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Or it could be something
like five over x plus y

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is equal to ten.

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This also is not going to be a line.

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So now, and at some point you could--

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I encourage you to try
to graph these things,

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they're actually quite interesting.

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But given that we've now seen
examples of linear equations

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and non-linear equations,

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let's see if we can come
up with a definition for

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linear equations.

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One way to think about is it's an equation

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that if you were to graph
all of the x and y pairs that

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satisfy this equation,

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you'll get a line.

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And that's actually
literally where the word

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linear equation comes from.

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But another way to think about it is

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it's going to be an equation where

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every term is either
going to be a constant,

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so for example, twelve is a constant.

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It's not going to change
based on the value of

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some variable, twelve is twelve.

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Or negative three is negative three.

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So every term is either
going to be a constant

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or it's going to be a
constant times a variable

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raised to the first power.

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So this is the constant two
times x to the first power.

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This is the variable y
raised to the first power.

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You could say that bceause
this is just one y.

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We're not dividing by x or y,

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we're not multiplying,

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we don't have a term that
has x to the second power,

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or x to the third power,
or y to the fifth power.

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We just have y to the first power,

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we have x to the first power.

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We're not multiplying x and y together

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like we did over here.

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So if every--

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if every term in your equation,

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on either side of the equation,

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is either a constant or its
just some number times x,

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just x to the first power
or some number times y,

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and you're not multiplying
your x's and y's together

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you are dealing with a linear equation.


Mistakes in solving equations

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Kind: captions
Language: en

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- [Voiceover] What I wanna do in this video

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is get some practice building an important skill,

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being able to look at some algebra that has happened

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and see whether it is correct.

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And this is valuable.

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You might look at other people's' work,

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you might look at your own work and be able to tell,

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"Well, hey, did I do that right?"

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The logic maybe broke down.

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Or why am I not getting the answer

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that I think I should get?

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So let's get some practice here.

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So I have an equation,

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nine times, and then in parentheses,

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two x plus 1/3 is equal to 39.

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And then someone has worked it out.

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This is step one, step two, then step three.

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They get to this solution, x equals two.

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What I want you to do is pause this video

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and figure out, is this solution correct?

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Did all of the steps they do, do they make logical sense,

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do they make algebraic sense?

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Or did they make a mistake in one of these steps,

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and if so, which of these steps did they make a mistake in?

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All right, so let's just work through this problem

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and see if all of these steps are reasonable.

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So in this first step, let's see, they got an 18x here,

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so it looks like they distributed the nine.

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So nine times two x would be 18x, and then of course,

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you would also have to do nine times 1/3.

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Nine times 1/3 would be 9/3, or three,

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so that seems reasonable.

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And you still have a 39 here on the right-hand side.

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So step one checks out.

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Now, let's see, what did they do over here?

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After step one, or at this stage,

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you had 18x plus three,

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and then over here you only have 18x.

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So they must've subtracted three

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from the left-hand side to get rid of this three.

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So they subtracted three from the left-hand side.

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Well, if they subtract three from the left-hand side,

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they also need to subtract three from the right-hand side,

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and it looks like they did.

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39 minus three is 36. So that step checks out.

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And so you have 18x is equal to 36.

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And so to solve for x,

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you would just wanna divide both sides by 18.

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So you would get x is equal to 36 over 18, or 2.

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So all of these steps check out,

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and so there's no problem here, no mistake.

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And you can verify that x does indeed equal two.

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Let's do some more questions,

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or see if we can identify more errors or maybe no errors.

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All right, we have 8/3 is equal to three times c plus 5/3,

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and we wanna solve for c.

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Pause the video and see if any of these mistakes are,

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or if any of these steps, are incorrect

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or maybe there's no mistake at all.

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All right, let's go step by step.

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So in step one, kept the left-hand side the same.

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The right-hand side, let's see,

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you go from three times this to three c,

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so it looks like they're distributing the three.

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So three times c is indeed three c,

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and then you wanna distribute three times 5/3,

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well, that's gonna be five.

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That's not gonna be 5/3.

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This thing right over here, this thing needs to be five,

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not 5/3, so that is not correct.

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So step one is where we have the problem.

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Let's keep going. Let's do one more of these.

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I find this strangely entertaining.

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All right, this is a little bit simpler.

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1/4 r plus two is equal to 10.

00:03:09.273 --> 00:03:11.880
And then they immediately go from that to just 1/4 r,

00:03:11.880 --> 00:03:14.318
so they got rid of this two on the left-hand side.

00:03:14.318 --> 00:03:17.011
Well, to do that you could subtract two from the left,

00:03:17.011 --> 00:03:18.312
but, of course, you would also have to do that

00:03:18.312 --> 00:03:20.564
from the right-hand side.

00:03:20.564 --> 00:03:22.746
And so you get 1/4 r plus two minus two,

00:03:22.746 --> 00:03:24.093
that would be just 1/4 r.

00:03:24.093 --> 00:03:26.021
So the left-hand side makes sense.

00:03:26.021 --> 00:03:29.551
10 minus two is not 12.

00:03:29.551 --> 00:03:32.081
10 minus two is eight.

00:03:32.081 --> 00:03:33.404
This right over here should be eight.

00:03:33.404 --> 00:03:34.937
They added two on the right-hand side.

00:03:34.937 --> 00:03:36.957
They subtracted two on the left-hand side,

00:03:36.957 --> 00:03:39.023
and they added two on the right-hand side.

00:03:39.023 --> 00:03:41.020
Then the equality wouldn't hold anymore.

00:03:41.020 --> 00:03:42.692
So you definitely have an error.

00:03:42.692 --> 00:03:45.269
Step one is not correct.


Manipulating formula example

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:03.150
The equation y is equal
to the square root of 2x

00:00:03.150 --> 00:00:06.110
minus 10 has a y-intercept
of 0, negative 10,

00:00:06.110 --> 00:00:08.340
and you see that
from this equation.

00:00:08.340 --> 00:00:11.520
The y-intercept is where do
you intersect the y-axis,

00:00:11.520 --> 00:00:15.140
or what happens to y
when x is equal to 0.

00:00:15.140 --> 00:00:17.680
When x is equal to 0, this
part of the expression

00:00:17.680 --> 00:00:22.570
is equal to 0, and you're left
with y is equal to negative 10.

00:00:22.570 --> 00:00:26.870
Rearrange the equation to
express x in terms of y.

00:00:26.870 --> 00:00:28.630
So I've got to put
this on my scratch pad

00:00:28.630 --> 00:00:31.080
here, so that I can work it out.

00:00:31.080 --> 00:00:33.980
So really, we just want
to think about-- we just

00:00:33.980 --> 00:00:35.920
want to solve for x here.

00:00:35.920 --> 00:00:38.340
We want to rearrange to
express x in terms of y,

00:00:38.340 --> 00:00:39.750
so let me just rewrite it.

00:00:39.750 --> 00:00:45.080
y is equal to the square
root of 2x minus 10.

00:00:45.080 --> 00:00:47.630
So we want to isolate the
x on the right-hand side,

00:00:47.630 --> 00:00:49.213
so the first thing
we might want to do

00:00:49.213 --> 00:00:50.740
is just add 10 to both sides.

00:00:50.740 --> 00:00:53.510
Let me add 10 to both sides.

00:00:53.510 --> 00:00:55.880
That gets rid of the 10
on the right-hand side.

00:00:55.880 --> 00:01:00.030
And we are left with
y plus 10-- Actually

00:01:00.030 --> 00:01:02.230
let me just keep
track of the colors

00:01:02.230 --> 00:01:07.690
just so you can see
what I'm up to-- is

00:01:07.690 --> 00:01:12.500
equal to the square root of 2x.

00:01:12.500 --> 00:01:14.320
Now, let's see.

00:01:14.320 --> 00:01:17.430
To isolate this x, let's square
both sides here, so let's

00:01:17.430 --> 00:01:18.810
square the left-hand side.

00:01:18.810 --> 00:01:20.300
Let's square the
right-hand side.

00:01:20.300 --> 00:01:22.925
Of course, anything we do to one
side, we have to do the other,

00:01:22.925 --> 00:01:25.080
otherwise the equality
will no longer be true.

00:01:25.080 --> 00:01:34.060
And we are going to
have y plus 10 squared

00:01:34.060 --> 00:01:38.050
is equal to-- the square
root of 2x squared

00:01:38.050 --> 00:01:41.220
is just going to be
2x-- is equal to 2x.

00:01:41.220 --> 00:01:42.990
And now we can divide
both sides by 2.

00:01:46.330 --> 00:01:53.070
And we are left with x is equal
to y plus 10 squared over 2.

00:01:53.070 --> 00:01:55.430
So let me write that out.

00:01:55.430 --> 00:01:58.370
Let me actually type that
in into the program right

00:01:58.370 --> 00:01:58.870
over here.

00:01:58.870 --> 00:02:07.040
So x was equal to
y plus 10 squared

00:02:07.040 --> 00:02:11.740
over 2, which is exactly what
I wrote down on my scratch pad.

00:02:11.740 --> 00:02:14.590
And then they say what is
the equation's x-intercept?

00:02:14.590 --> 00:02:17.720
So let's just remind ourselves
what's going on here.

00:02:17.720 --> 00:02:20.610
So this curve or
this equation could

00:02:20.610 --> 00:02:22.620
describe a curve that
looks something like this.

00:02:22.620 --> 00:02:24.660
So that's my y-axis.

00:02:24.660 --> 00:02:27.930
This is my x-axis.

00:02:27.930 --> 00:02:30.585
And let's say the curve, it
might look something like this.

00:02:33.290 --> 00:02:35.970
It might look
something like that.

00:02:35.970 --> 00:02:37.790
So we already figured
out its y-intercept.

00:02:37.790 --> 00:02:40.510
Its y-intercept is where
does it intersect the y-axis,

00:02:40.510 --> 00:02:42.930
and we already figured out
that's when x is equal to 0,

00:02:42.930 --> 00:02:45.250
y is equal to negative 10.

00:02:45.250 --> 00:02:47.780
Now we want to figure
out its x-intercept,

00:02:47.780 --> 00:02:49.680
so that's essentially
what x value

00:02:49.680 --> 00:02:52.850
do we get to when
y is equal to 0?

00:02:52.850 --> 00:02:56.090
So what x value do we get
to when y is equal to 0?

00:02:56.090 --> 00:02:59.880
Let me just-- let me clear these
out just so this doesn't-- we

00:02:59.880 --> 00:03:02.130
don't get distracted by
this right over here.

00:03:02.130 --> 00:03:08.140
Well, when y is equal
to 0, x is going

00:03:08.140 --> 00:03:13.000
to be equal to 10
squared over 2.

00:03:13.000 --> 00:03:16.350
0 plus 10 is 10
squared, 100/2, which

00:03:16.350 --> 00:03:18.590
is going to be equal to 50.

00:03:18.590 --> 00:03:21.990
So we're at the point
x is 50, and y is 0.

00:03:21.990 --> 00:03:24.250
So that's this right over here.

00:03:24.250 --> 00:03:26.220
The x-intercept is 50.

00:03:26.220 --> 00:03:29.550
When x is 50, we clearly
see that y is equal to 0.

00:03:29.550 --> 00:03:31.480
So let's also put that in there.

00:03:31.480 --> 00:03:33.830
So what is the
equation's x-intercept?

00:03:33.830 --> 00:03:39.110
So when x is 50, y is 0.

00:03:39.110 --> 00:03:42.230
And we can check our
answer, and we got it right.


Number of solutions to linear equations ex 2

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:02.700
We're asked to use
the drop-downs to form

00:00:02.700 --> 00:00:06.330
a linear equation
with no solutions.

00:00:06.330 --> 00:00:08.990
So a linear equation
with no solutions

00:00:08.990 --> 00:00:11.520
is going to be one where I don't
care how you manipulate it,

00:00:11.520 --> 00:00:13.160
the thing on the
left can never be

00:00:13.160 --> 00:00:14.494
equal to the thing on the right.

00:00:14.494 --> 00:00:16.284
And so let's see what
options they give us.

00:00:16.284 --> 00:00:18.440
One, they want us to-- we
can pick the coefficient

00:00:18.440 --> 00:00:21.840
on the x term and then
we can pick the constant.

00:00:21.840 --> 00:00:24.000
So if we made this
negative 11x, so now

00:00:24.000 --> 00:00:26.420
we have a negative
11x on both sides.

00:00:26.420 --> 00:00:29.380
Here on the left hand side,
we have negative 11x plus 4.

00:00:29.380 --> 00:00:32.159
If we do something other
than 4 here, so if we did

00:00:32.159 --> 00:00:36.160
say negative 11x minus
11, then here we're

00:00:36.160 --> 00:00:37.810
not going to have any solutions.

00:00:37.810 --> 00:00:40.120
And you say, hey, Sal how
did you come up with that?

00:00:40.120 --> 00:00:41.620
Well think about
it right over here.

00:00:41.620 --> 00:00:45.109
We have a negative 11x here,
we have a negative 11x there.

00:00:45.109 --> 00:00:46.900
If you wanted to solve
it algebraically you

00:00:46.900 --> 00:00:49.550
could add 11x to both sides
and both of these terms

00:00:49.550 --> 00:00:52.430
will cancel out with each other
and all you would be left with

00:00:52.430 --> 00:00:55.010
is a 4 is equal to
a negative 11, which

00:00:55.010 --> 00:00:57.840
is not possible for
any x that you pick.

00:00:57.840 --> 00:00:59.920
Another way that you
think about it is here we

00:00:59.920 --> 00:01:01.880
have negative 11
times some number

00:01:01.880 --> 00:01:04.005
and we're adding 4
to it, and here we're

00:01:04.005 --> 00:01:06.130
taking negative 11 times
that same number and we're

00:01:06.130 --> 00:01:07.860
subtracting 11 from it.

00:01:07.860 --> 00:01:09.850
So if you take a negative
11 times some number

00:01:09.850 --> 00:01:12.820
and on one side you add four,
and on the other side you

00:01:12.820 --> 00:01:16.120
subtract 11, there's no way, it
doesn't matter what x you pick.

00:01:16.120 --> 00:01:19.020
There's no x for which
that is going to be true.

00:01:19.020 --> 00:01:22.830
But let's check our
answer right over here.


Number of solutions to linear equations ex 3

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:03.440
We're asked to use the drop-down
to form a linear equation

00:00:03.440 --> 00:00:06.870
with infinitely many solutions.

00:00:06.870 --> 00:00:08.940
So an equation with
infinitely many solutions

00:00:08.940 --> 00:00:11.120
essentially has the same
thing on both sides,

00:00:11.120 --> 00:00:13.350
no matter what x you pick.

00:00:13.350 --> 00:00:16.789
So first, my brain just wants
to simplify this left-hand side

00:00:16.789 --> 00:00:18.330
a little bit and
then think about how

00:00:18.330 --> 00:00:19.804
I can engineer the
right-hand side

00:00:19.804 --> 00:00:21.470
so it's going to be
the same as the left

00:00:21.470 --> 00:00:23.420
no matter what x I pick.

00:00:23.420 --> 00:00:24.950
So right over here,
if I distribute

00:00:24.950 --> 00:00:29.520
the 4 over x minus
2, I get 4x minus 8.

00:00:29.520 --> 00:00:31.820
And then I'm adding x to that.

00:00:31.820 --> 00:00:35.360
And that's, of course, going
to be equal to 5x plus blank.

00:00:35.360 --> 00:00:38.340
And I get to pick
what my blank is.

00:00:38.340 --> 00:00:43.940
And so 4x plus x is 5x.

00:00:43.940 --> 00:00:46.240
And of course, we
still have our minus 8.

00:00:46.240 --> 00:00:51.150
And that's going to be
equal to 5x plus blank.

00:00:51.150 --> 00:00:52.840
So what could we make
that blank so this

00:00:52.840 --> 00:00:55.770
is true for any x we pick?

00:00:55.770 --> 00:00:59.430
Well, over here we have
5 times an x minus 8.

00:00:59.430 --> 00:01:03.109
Well, if we make this a minus
8, or if we subtract 8 here,

00:01:03.109 --> 00:01:04.650
or if we make this
a negative 8, this

00:01:04.650 --> 00:01:06.710
is going to be true for any x.

00:01:06.710 --> 00:01:08.970
So if we make this
a negative 8, this

00:01:08.970 --> 00:01:10.750
is going to be true
for any x you pick.

00:01:10.750 --> 00:01:13.770
You give me any x, you multiply
it by 5 and subtract 8,

00:01:13.770 --> 00:01:16.790
that's, of course, going to
be that same x multiplied by 5

00:01:16.790 --> 00:01:18.210
and subtracting 8.

00:01:18.210 --> 00:01:20.460
And if you were to try to
somehow solve this equation,

00:01:20.460 --> 00:01:23.660
subtract 5x from both sides,
you would get negative 8

00:01:23.660 --> 00:01:26.960
is equal to negative 8, which is
absolutely true for absolutely

00:01:26.960 --> 00:01:28.650
any x that you pick.

00:01:28.650 --> 00:01:32.410
So let's go-- let me actually
fill this in on the exercise.

00:01:32.410 --> 00:01:38.090
So I want to make 5-- it's
going to be 5x plus negative 8.


Number of solutions to linear equations

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:02.950
Determine the
number of solutions

00:00:02.950 --> 00:00:05.320
for each of these
equations, and they give us

00:00:05.320 --> 00:00:07.270
three equations right over here.

00:00:07.270 --> 00:00:09.730
And before I deal with these
equations in particular,

00:00:09.730 --> 00:00:12.070
let's just remind
ourselves about when

00:00:12.070 --> 00:00:16.640
we might have one or
infinite or no solutions.

00:00:16.640 --> 00:00:18.200
You're going to
have one solution

00:00:18.200 --> 00:00:20.290
if you can, by
solving the equation,

00:00:20.290 --> 00:00:24.050
come up with something like
x is equal to some number.

00:00:24.050 --> 00:00:25.600
Let's say x is
equal to-- if I want

00:00:25.600 --> 00:00:27.460
to say the abstract--
x is equal to a.

00:00:27.460 --> 00:00:29.060
Or if we actually
were to solve it,

00:00:29.060 --> 00:00:32.285
we'd get something like x
equals 5 or 10 or negative pi--

00:00:32.285 --> 00:00:33.160
whatever it might be.

00:00:33.160 --> 00:00:35.860
But if you could actually
solve for a specific x,

00:00:35.860 --> 00:00:37.730
then you have one solution.

00:00:37.730 --> 00:00:41.470
So this is one solution,
just like that.

00:00:41.470 --> 00:00:44.380
Now if you go and you try to
manipulate these equations

00:00:44.380 --> 00:00:47.400
in completely legitimate
ways, but you end up

00:00:47.400 --> 00:00:52.380
with something crazy
like 3 equals 5,

00:00:52.380 --> 00:00:55.470
then you have no solutions.

00:00:55.470 --> 00:00:57.320
And if you just think
about it reasonably,

00:00:57.320 --> 00:01:00.770
all of these equations
are about finding

00:01:00.770 --> 00:01:02.617
an x that satisfies this.

00:01:02.617 --> 00:01:04.450
And if you were to just
keep simplifying it,

00:01:04.450 --> 00:01:07.030
and you were to get
something like 3 equals 5,

00:01:07.030 --> 00:01:08.950
and you were to ask
yourself the question

00:01:08.950 --> 00:01:14.210
is there any x that can somehow
magically make 3 equal 5, no.

00:01:14.210 --> 00:01:16.900
No x can magically
make 3 equal 5,

00:01:16.900 --> 00:01:20.240
so there's no way that you could
make this thing be actually

00:01:20.240 --> 00:01:21.762
true, no matter
which x you pick.

00:01:21.762 --> 00:01:23.720
So if you get something
very strange like this,

00:01:23.720 --> 00:01:25.490
this means there's no solution.

00:01:25.490 --> 00:01:28.779
On the other hand, if you get
something like 5 equals 5--

00:01:28.779 --> 00:01:30.320
and I'm just over
using the number 5.

00:01:30.320 --> 00:01:31.736
It didn't have to
be the number 5.

00:01:31.736 --> 00:01:34.802
It could be 7 or 10
or 113, whatever.

00:01:34.802 --> 00:01:36.260
And actually let
me just not use 5,

00:01:36.260 --> 00:01:38.886
just to make sure that you
don't think it's only for 5.

00:01:38.886 --> 00:01:40.510
If I just get something,
that something

00:01:40.510 --> 00:01:42.350
is equal to itself,
which is just

00:01:42.350 --> 00:01:46.980
going to be true no matter what
x you pick, any x you pick,

00:01:46.980 --> 00:01:48.340
this would be true for.

00:01:48.340 --> 00:01:52.690
Well, then you have
an infinite solutions.

00:01:52.690 --> 00:01:54.660
So with that as a
little bit of a primer,

00:01:54.660 --> 00:01:57.890
let's try to tackle
these three equations.

00:01:57.890 --> 00:01:59.130
So over here, let's see.

00:01:59.130 --> 00:02:00.500
Maybe we could subtract.

00:02:00.500 --> 00:02:03.090
If we want to get rid of this
2 here on the left hand side,

00:02:03.090 --> 00:02:06.080
we could subtract
2 from both sides.

00:02:06.080 --> 00:02:08.741
If we subtract 2
from both sides,

00:02:08.741 --> 00:02:10.699
we are going to be left
with-- on the left hand

00:02:10.699 --> 00:02:13.864
side we're going to be
left with negative 7x.

00:02:13.864 --> 00:02:15.280
And on the right
hand side, you're

00:02:15.280 --> 00:02:17.270
going to be left with 2x.

00:02:17.270 --> 00:02:20.410
This is going to
cancel minus 9x.

00:02:20.410 --> 00:02:23.730
2x minus 9x, If we simplify
that, that's negative 7x.

00:02:23.730 --> 00:02:27.220
You get negative 7x is
equal to negative 7x.

00:02:27.220 --> 00:02:29.060
And you probably see
where this is going.

00:02:29.060 --> 00:02:32.380
This is already true
for any x that you pick.

00:02:32.380 --> 00:02:36.620
Negative 7 times that x is going
to be equal to negative 7 times

00:02:36.620 --> 00:02:37.740
that x.

00:02:37.740 --> 00:02:40.120
So we already are going
into this scenario.

00:02:40.120 --> 00:02:42.330
But you're like hey, so
I don't see 13 equals 13.

00:02:42.330 --> 00:02:44.880
Well, what if you did
something like you

00:02:44.880 --> 00:02:48.812
divide both sides by negative 7.

00:02:48.812 --> 00:02:51.020
At this point, what I'm
doing is kind of unnecessary.

00:02:51.020 --> 00:02:53.353
You already understand that
negative 7 times some number

00:02:53.353 --> 00:02:55.910
is always going to be
negative 7 times that number.

00:02:55.910 --> 00:03:00.610
But if we were to do this,
we would get x is equal to x,

00:03:00.610 --> 00:03:03.410
and then we could subtract
x from both sides.

00:03:03.410 --> 00:03:05.420
And then you would
get zero equals

00:03:05.420 --> 00:03:09.010
zero, which is true for
any x that you pick.

00:03:09.010 --> 00:03:10.880
Zero is always going
to be equal to zero.

00:03:10.880 --> 00:03:12.379
So any of these
statements are going

00:03:12.379 --> 00:03:13.850
to be true for any x you pick.

00:03:13.850 --> 00:03:16.360
So for this equation
right over here,

00:03:16.360 --> 00:03:21.850
we have an infinite
number of solutions.

00:03:21.850 --> 00:03:24.520
Let's think about this one
right over here in the middle.

00:03:24.520 --> 00:03:26.144
So once again, let's try it.

00:03:26.144 --> 00:03:27.560
I'll do it a little
bit different.

00:03:27.560 --> 00:03:32.080
I'll add this 2x and this
negative 9x right over there.

00:03:32.080 --> 00:03:40.760
So we will get negative 7x
plus 3 is equal to negative 7x.

00:03:40.760 --> 00:03:46.860
So 2x plus 9x is
negative 7x plus 2.

00:03:46.860 --> 00:03:50.730
Well, let's add-- why don't we
do that in that green color.

00:03:50.730 --> 00:03:52.400
Let's do that in
that green color.

00:03:52.400 --> 00:03:55.090
Plus 2, this is 2.

00:03:55.090 --> 00:03:56.830
Now let's add 7x to both sides.

00:03:56.830 --> 00:03:58.907
Well if you add 7x to
the left hand side,

00:03:58.907 --> 00:04:00.740
you're just going to
be left with a 3 there.

00:04:00.740 --> 00:04:02.764
And if you add 7x to
the right hand side,

00:04:02.764 --> 00:04:04.430
this is going to go
away and you're just

00:04:04.430 --> 00:04:05.763
going to be left with a 2 there.

00:04:05.763 --> 00:04:08.810
So all I did is I added 7x.

00:04:08.810 --> 00:04:12.660
I added 7x to both
sides of that equation.

00:04:12.660 --> 00:04:14.750
And now we've got
something nonsensical.

00:04:14.750 --> 00:04:18.320
I don't care what x you pick,
how magical that x might be.

00:04:18.320 --> 00:04:22.320
There's no way that that x is
going to make 3 equal to 2.

00:04:22.320 --> 00:04:29.100
So in this scenario right over
here, we have no solutions.

00:04:29.100 --> 00:04:34.630
There's no x in the universe
that can satisfy this equation.

00:04:34.630 --> 00:04:37.300
Now let's try this
third scenario.

00:04:37.300 --> 00:04:41.200
So once again, maybe we'll
subtract 3 from both sides,

00:04:41.200 --> 00:04:43.390
just to get rid of
this constant term.

00:04:43.390 --> 00:04:46.910
So we're going to get negative
7x on the left hand side.

00:04:46.910 --> 00:04:49.883
On the right hand side, we're
going to have 2x minus 1.

00:04:52.470 --> 00:04:56.870
And now we can subtract
2x from both sides.

00:04:56.870 --> 00:04:58.930
To subtract 2x from
both sides, you're

00:04:58.930 --> 00:05:02.280
going to get-- so
subtracting 2x,

00:05:02.280 --> 00:05:06.320
you're going to get negative
9x is equal to negative 1.

00:05:06.320 --> 00:05:09.970
Now you can divide both
sides by negative 9.

00:05:09.970 --> 00:05:13.320
And you are left with
x is equal to 1/9.

00:05:13.320 --> 00:05:15.270
So we're in this
scenario right over here.

00:05:15.270 --> 00:05:17.625
We very explicitly
were able to find an x,

00:05:17.625 --> 00:05:21.220
x equals 1/9, that
satisfies this equation.

00:05:21.220 --> 00:05:26.920
So this right over here
has exactly one solution.


Figuring out missing algebraic step

WEBVTT
Kind: captions
Language: en

00:00:00.727 --> 00:00:02.584
Voiceover:Below are the
steps that Steven used

00:00:02.584 --> 00:00:06.248
to correctly solve the
equation six X is equal to

00:00:06.248 --> 00:00:09.926
three times X plus four, four X.

00:00:09.926 --> 00:00:11.990
Step one is missing.

00:00:11.990 --> 00:00:15.131
What could Steven have
written for step one?

00:00:15.131 --> 00:00:17.643
So let's see, so this is
the original equation,

00:00:17.643 --> 00:00:19.643
they give us a blank for step one,

00:00:19.643 --> 00:00:22.075
and then we see that he eventually
gets to three X equals 12.

00:00:22.075 --> 00:00:24.316
That he divides both sides by three

00:00:24.316 --> 00:00:25.804
and he gets to X equals four.

00:00:25.804 --> 00:00:28.742
So the best way I could
think about doing this

00:00:28.742 --> 00:00:30.500
is to think, what would I do?

00:00:30.500 --> 00:00:31.845
Well if I saw something like this,

00:00:31.845 --> 00:00:34.948
my extinct would be to
distribute this three

00:00:34.948 --> 00:00:39.265
so that I could start separating the parts

00:00:39.265 --> 00:00:40.545
of this expression that have a variable,

00:00:40.545 --> 00:00:42.816
that have the X, and the
parts that are constant.

00:00:42.816 --> 00:00:46.113
So let's try to distribute
this three and see what we get.

00:00:46.113 --> 00:00:48.272
and see if that's a reasonable step for

00:00:48.272 --> 00:00:50.416
the direction that Steven went in.

00:00:50.416 --> 00:00:53.360
So if we do that, we'd
get six X is equal to,

00:00:53.360 --> 00:00:54.736
we haven't changed the left-hand side,

00:00:54.736 --> 00:00:58.188
and the right-hand side
three times X is three X.

00:00:58.188 --> 00:01:01.502
Then three times four is 12.

00:01:01.502 --> 00:01:05.291
So you get six X is
equal to three X plus 12.

00:01:05.291 --> 00:01:08.098
Now if I don't even look
at what Steven's doing

00:01:08.098 --> 00:01:10.866
right over here, if I'm
trying to solve for X,

00:01:10.866 --> 00:01:12.529
what would be the next
thing that I would do?

00:01:12.529 --> 00:01:15.505
Well, I'd try to isolate
all of the X's on one side,

00:01:15.505 --> 00:01:18.202
and if I want all the
X's on the left-hand side

00:01:18.202 --> 00:01:21.618
I would subtract three X from both sides.

00:01:21.618 --> 00:01:28.599
So if you subtract three X from both sides

00:01:28.661 --> 00:01:32.280
you would then get Steven's
second step right over here.

00:01:32.280 --> 00:01:35.735
So this looks like a completely

00:01:35.735 --> 00:01:38.237
reasonable first step for Steven.

00:01:38.237 --> 00:01:40.589
The first step is he distributes the three

00:01:40.589 --> 00:01:43.773
to get this expression,
then he subtracts three X

00:01:43.773 --> 00:01:46.830
from both sides to get
three X is equal to 12,

00:01:46.830 --> 00:01:49.646
and then he divides both sides by three

00:01:49.646 --> 00:01:52.384
to get X is equal to four.


Understanding steps when solving equations

WEBVTT
Kind: captions
Language: en

00:00:00.466 --> 00:00:03.246
Voiceover:Create a list
of steps, in order,

00:00:03.246 --> 00:00:05.433
that will solve the following equation.

00:00:05.433 --> 00:00:06.439
It gives the equation here

00:00:06.439 --> 00:00:07.850
then they have a bunch of steps

00:00:07.850 --> 00:00:09.298
that they want us to put in order.

00:00:09.298 --> 00:00:11.642
Let's just work it out
on this scratch pad.

00:00:11.642 --> 00:00:12.864
You might be able to work this out

00:00:12.864 --> 00:00:13.770
in your head eventually

00:00:13.770 --> 00:00:15.661
but I'll do it on this scratch pad first.

00:00:15.661 --> 00:00:17.739
First, I'm just going to solve this

00:00:17.739 --> 00:00:19.602
the way that I might try to solve it.

00:00:19.602 --> 00:00:24.624
Let's say we have 5x
minus 11 is equal to 42

00:00:24.624 --> 00:00:26.487
and as we've seen before

00:00:26.487 --> 00:00:28.060
whenever I like to solve something,

00:00:28.060 --> 00:00:29.474
I like to isolate the variable

00:00:29.474 --> 00:00:30.786
that I'm trying to solve for.

00:00:30.786 --> 00:00:33.228
I want to isolate this 5x on one side

00:00:33.228 --> 00:00:34.743
and then eventually I'm
going to try to get that

00:00:34.743 --> 00:00:35.812
to being an x.

00:00:35.812 --> 00:00:38.023
The best way to isolate it

00:00:38.023 --> 00:00:40.004
is to get rid of this minus 11.

00:00:40.004 --> 00:00:41.639
The best way to get this minus,

00:00:41.639 --> 00:00:43.933
get rid of that minus
11 on the left hand side

00:00:43.933 --> 00:00:46.310
is to add 11 to the left hand side

00:00:46.310 --> 00:00:48.050
but I can't just do it
only to the left hand side,

00:00:48.050 --> 00:00:50.262
then these two things
won't be equal anymore.

00:00:50.262 --> 00:00:52.072
In order for them to both be equal,

00:00:52.072 --> 00:00:54.916
I have to do the same thing to both sides.

00:00:54.916 --> 00:01:00.102
Then I would be left with
well, negative 11 plus 11

00:01:00.102 --> 00:01:01.181
is going to be zero.

00:01:01.181 --> 00:01:06.329
I'm going to be left
with 5x is equal to 53.

00:01:06.329 --> 00:01:09.928
What was the first step that I just did?

00:01:09.928 --> 00:01:12.140
Well I added 11 to both sides,

00:01:12.140 --> 00:01:15.891
so this is going to be my first step.

00:01:15.891 --> 00:01:19.222
Now what do I do to solve this.

00:01:19.222 --> 00:01:21.301
Once again I want this to be on the form

00:01:21.301 --> 00:01:22.464
of x equals something

00:01:22.464 --> 00:01:23.998
that I know what x is.

00:01:23.998 --> 00:01:26.060
What's the best way to just have an x here

00:01:26.060 --> 00:01:27.489
on the left hand side?

00:01:27.489 --> 00:01:29.721
I could divide 5x by five

00:01:29.721 --> 00:01:30.874
that would give me x.

00:01:30.874 --> 00:01:32.714
Once again if I want these two things

00:01:32.714 --> 00:01:34.527
to be equal to each other,

00:01:34.527 --> 00:01:36.804
I have to do the same thing to both sides.

00:01:36.804 --> 00:01:38.992
I have to divide 53 by five as well

00:01:38.992 --> 00:01:40.639
and then I'll be left with,

00:01:40.639 --> 00:01:47.414
I'll be left with x is
equal to 53 over five

00:01:47.414 --> 00:01:49.096
and I am done.

00:01:49.096 --> 00:01:51.597
I have solved for the x that
satisfies this equation.

00:01:51.597 --> 00:01:53.209
What's the next thing that I did?

00:01:53.209 --> 00:01:55.017
I divided both sides by five,

00:01:55.017 --> 00:01:56.619
so this right over here is step two.

00:01:56.619 --> 00:01:57.860
I added 11 to both sides

00:01:57.860 --> 00:01:59.582
then divide both sides by five.

00:01:59.582 --> 00:02:00.734
Let's fill that in.

00:02:00.734 --> 00:02:04.374
I'm going to add 11 to both sides,

00:02:04.374 --> 00:02:05.572
adding 11 to both sides

00:02:05.572 --> 00:02:08.027
then I'm just left with 5x is equal to 53

00:02:08.027 --> 00:02:11.806
and then I divide both sides by five.

00:02:13.498 --> 00:02:14.434
Got it right.


How to solve two-step equations with decimals and fractions

WEBVTT
Kind: captions
Language: en

00:00:00.225 --> 00:00:02.024
- Let's get some practice solving equations

00:00:02.024 --> 00:00:04.044
that involve fractions and decimals.

00:00:04.044 --> 00:00:06.703
And this equation clearly involves fractions.

00:00:06.703 --> 00:00:07.307
Let's see.

00:00:07.307 --> 00:00:11.173
We have negative 1/3 = j/4 - 10/3.

00:00:11.173 --> 00:00:12.252
So I encourage you to pause the video

00:00:12.252 --> 00:00:13.494
and see if you could solve for j.

00:00:13.494 --> 00:00:16.849
What j would make this equation true?

00:00:16.849 --> 00:00:17.650
Alright.

00:00:17.650 --> 00:00:19.148
Now let's work through this together.

00:00:19.148 --> 00:00:20.994
So what I like to do, I like to isolate

00:00:20.994 --> 00:00:22.050
the variable that I'm trying to solve for

00:00:22.050 --> 00:00:23.617
on one side.

00:00:23.617 --> 00:00:25.196
Since it's already on the right-hand side,

00:00:25.196 --> 00:00:26.589
let's try to get all the things that involve

00:00:26.589 --> 00:00:28.191
j on the right-hand side, and then

00:00:28.191 --> 00:00:29.689
get rid of everything else on the

00:00:29.689 --> 00:00:30.908
right-hand side.

00:00:30.908 --> 00:00:32.835
So I want to get rid of this negative 10/3,

00:00:32.835 --> 00:00:34.553
and the best way I can think of doing that

00:00:34.553 --> 00:00:35.992
is by adding 10/3.

00:00:35.992 --> 00:00:38.349
Now I can't just do that to one side

00:00:38.349 --> 00:00:39.011
of the equation.

00:00:39.011 --> 00:00:40.195
Then it wouldn't be equal anymore.

00:00:40.195 --> 00:00:42.052
If this is equal to that, in order for the

00:00:42.052 --> 00:00:43.712
quantity to be true, whatever I do to this

00:00:43.712 --> 00:00:45.500
I have to do to that as well.

00:00:45.500 --> 00:00:48.240
So I have to add 10/3 to both sides.

00:00:48.240 --> 00:00:49.761
I have to add 10/3 to both sides

00:00:49.761 --> 00:00:50.399
of the equation.

00:00:50.399 --> 00:00:52.334
And so what am I going to get?

00:00:52.334 --> 00:00:54.656
On the left-hand side,

00:00:54.656 --> 00:00:57.117
I'm going to have negative 1/3 + 10/3,

00:00:57.117 --> 00:00:58.835
which is 9/3.

00:00:58.835 --> 00:01:00.518
9/3.

00:01:00.518 --> 00:01:02.875
And then that's going to be equal to.

00:01:02.875 --> 00:01:04.442
And on the right-hand side,

00:01:04.442 --> 00:01:07.135
the negative 10/3 and the positive 10/3,

00:01:07.135 --> 00:01:09.062
those cancel out to get a zero,

00:01:09.062 --> 00:01:10.804
and I'm just left with j/4.

00:01:10.804 --> 00:01:13.718
It's equal to j/4.

00:01:13.718 --> 00:01:16.295
Now you might recognize 9/3, that's the

00:01:16.295 --> 00:01:18.059
same thing as nine divided by three.

00:01:18.059 --> 00:01:20.579
So this is just going to be three.

00:01:20.579 --> 00:01:21.960
So that simplifies a little bit.

00:01:21.960 --> 00:01:23.063
Three, let me just rewrite it so

00:01:23.063 --> 00:01:24.665
you don't get confused.

00:01:24.665 --> 00:01:28.856
Three is equal to j/4.

00:01:28.856 --> 00:01:31.317
Now, to solve for j,

00:01:31.317 --> 00:01:33.418
I could just multiply both sides by four.

00:01:33.418 --> 00:01:34.834
'Cause if I divide something by four

00:01:34.834 --> 00:01:36.274
and then multiply by four, I'm just

00:01:36.274 --> 00:01:37.435
going to be left with that something.

00:01:37.435 --> 00:01:40.558
If I start with j and I divide by four,

00:01:40.558 --> 00:01:42.682
and then I multiply, and then I

00:01:42.682 --> 00:01:44.798
multiply by four, so I'm just going to

00:01:44.798 --> 00:01:46.087
multiply by four, then I'm just going to be

00:01:46.087 --> 00:01:47.933
left with j on the right-hand side.

00:01:47.933 --> 00:01:48.769
But I can't just multiply the

00:01:48.769 --> 00:01:50.209
right-hand side by four.

00:01:50.209 --> 00:01:52.194
I have to do it with the left-hand side

00:01:52.194 --> 00:01:52.704
as well.

00:01:52.704 --> 00:01:53.528
So I multiply the left-hand side

00:01:53.528 --> 00:01:54.968
by four as well.

00:01:54.968 --> 00:01:56.710
And what I will be left with,

00:01:56.710 --> 00:01:59.332
four times three is twelve.

00:01:59.332 --> 00:02:01.469
And then j divided by four times four,

00:02:01.469 --> 00:02:03.710
well that's just going to be j.

00:02:03.710 --> 00:02:06.775
So we get j is equal to 12.

00:02:06.775 --> 00:02:08.667
And the neat thing about equations

00:02:08.667 --> 00:02:10.014
is you can verify that you indeed

00:02:10.014 --> 00:02:11.291
got the right answer.

00:02:11.291 --> 00:02:14.715
You can substitute 12 for j here,

00:02:14.715 --> 00:02:17.896
and verify that negative 1/3

00:02:17.896 --> 00:02:23.034
is equal to 12/4 - 10/3.

00:02:23.318 --> 00:02:24.339
Does this actually work out?

00:02:24.339 --> 00:02:27.879
Well 12/4 is the same thing as three,

00:02:27.879 --> 00:02:29.360
and if I wanted to write that as thirds,

00:02:29.360 --> 00:02:31.809
this is the same thing as 9/3.

00:02:31.809 --> 00:02:34.828
And 9/3 - 10/3 is indeed equal to

00:02:34.828 --> 00:02:36.685
negative 1/3.

00:02:36.685 --> 00:02:38.926
So we feel very good about that.

00:02:38.926 --> 00:02:41.407
Let's do another example.

00:02:41.407 --> 00:02:46.689
So I have n/5 + 0.6 = 2.

00:02:46.689 --> 00:02:49.788
So let's isolate this term

00:02:49.788 --> 00:02:51.147
that involves n on the left-hand side.

00:02:51.147 --> 00:02:53.248
So let's get rid of this 0.6.

00:02:53.248 --> 00:02:56.185
So let's subtract 0.6 from the left-hand side.

00:02:56.185 --> 00:02:57.207
But I can't just do it from the left.

00:02:57.207 --> 00:02:58.298
I have to do it from both sides

00:02:58.298 --> 00:03:00.261
if I want the equality to hold true.

00:03:00.261 --> 00:03:02.187
So, subtract 0.6.

00:03:02.187 --> 00:03:04.021
Now on the left-hand side,

00:03:04.021 --> 00:03:07.481
I'm just going to be left with n/5,

00:03:07.481 --> 00:03:09.419
and on the right-hand side,

00:03:09.419 --> 00:03:13.993
2 - 0.6, that;'s going to be 1.4.

00:03:13.993 --> 00:03:15.967
And if you don't want to do this

00:03:15.967 --> 00:03:17.209
in your head, you could work this out

00:03:17.209 --> 00:03:17.905
separately.

00:03:17.905 --> 00:03:22.375
It's going to be 2.0 - 0.6.

00:03:22.375 --> 00:03:26.090
You could say, "Oh, this is 20/10-6/10"

00:03:26.090 --> 00:03:28.493
which is going to be 14/10, which is that there.

00:03:28.493 --> 00:03:29.607
Or if you want to do it a little bit

00:03:29.607 --> 00:03:32.115
kind of the traditional method,

00:03:32.115 --> 00:03:32.939
you say, "Oh, I'm trying to subtract

00:03:32.939 --> 00:03:35.690
six from zero, let me re-group."

00:03:35.690 --> 00:03:37.153
That's going to be a 10.

00:03:37.153 --> 00:03:39.803
I'm going to take from the ones place.

00:03:39.803 --> 00:03:42.556
If I take a one from the one's place,

00:03:42.556 --> 00:03:44.263
and that's going to be equal to 10/10.

00:03:44.263 --> 00:03:46.956
10/10 - 6/10 is 4/10.

00:03:46.956 --> 00:03:49.552
And then, bring down one one

00:03:49.552 --> 00:03:51.315
minus zero ones is just one.

00:03:51.315 --> 00:03:52.338
So it's 1.4.

00:03:52.338 --> 00:03:53.592
And now, to solve for n.

00:03:53.592 --> 00:03:55.379
Well on the left have n being

00:03:55.379 --> 00:03:56.761
divided by five.

00:03:56.761 --> 00:03:57.631
If I just want n here, I can just

00:03:57.631 --> 00:03:59.314
multiply by five.

00:03:59.314 --> 00:04:01.937
So, if I multiply by five,

00:04:01.937 --> 00:04:05.073
five times n divided by five

00:04:05.073 --> 00:04:06.512
is going to be just n.

00:04:06.512 --> 00:04:08.172
But I can't just multiply the left-hand side

00:04:08.172 --> 00:04:08.659
by five.

00:04:08.659 --> 00:04:09.612
I have to multiply the right-hand side

00:04:09.612 --> 00:04:12.120
by five as well.

00:04:12.120 --> 00:04:14.093
And so what is that going to get us?

00:04:14.093 --> 00:04:18.550
We are going to get n = 1.4 x 5.

00:04:18.550 --> 00:04:22.114
1.4 x 5.

00:04:22.114 --> 00:04:24.007
Now you might be able to do this in your head,

00:04:24.007 --> 00:04:26.328
'cause this is one and 2/5.

00:04:26.328 --> 00:04:28.222
So this thing should all be equal to seven,

00:04:28.222 --> 00:04:29.824
but I'll just do it this way as well.

00:04:29.824 --> 00:04:33.736
Five times four is 20.

00:04:33.736 --> 00:04:35.117
Re-group the two.

00:04:35.117 --> 00:04:37.659
One times five is five plus two is seven.

00:04:37.659 --> 00:04:38.658
And when I look at all the numbers

00:04:38.658 --> 00:04:40.957
that I'm multiplying, I have one digit

00:04:40.957 --> 00:04:43.312
to the right of the decimal point.

00:04:43.312 --> 00:04:44.614
So my answer will have one digit

00:04:44.614 --> 00:04:45.972
to the right of the decimal point.

00:04:45.972 --> 00:04:48.954
So it's 7.0, or just 7.

00:04:48.954 --> 00:04:52.009
n = 7.

00:04:52.009 --> 00:04:53.820
And you can verify this works,

00:04:53.820 --> 00:04:55.600
'cause seven divided by five is

00:04:55.600 --> 00:04:59.076
going to be equal to 1.4, plus 0.6

00:04:59.076 --> 00:05:00.635
is equal to two.

00:05:00.635 --> 00:05:01.913
Let's do one more example.

00:05:01.913 --> 00:05:02.829
This is too much fun.

00:05:02.829 --> 00:05:03.537
Alright.

00:05:03.537 --> 00:05:06.811
0.5 times the whole quantity

00:05:06.811 --> 00:05:09.668
(r + 2.75) = 3.

00:05:09.668 --> 00:05:11.106
Now there's a bunch of ways

00:05:11.106 --> 00:05:12.187
that you could tackle this.

00:05:12.187 --> 00:05:13.347
A lot of times when you see

00:05:13.347 --> 00:05:14.416
something like this, your temptation

00:05:14.416 --> 00:05:17.249
might be, "Let's distribute the 0.5."

00:05:17.249 --> 00:05:18.966
But that makes it a little bit hairy,

00:05:18.966 --> 00:05:21.532
'cause 0.5 times 2.75.

00:05:21.532 --> 00:05:22.971
You could calculate that, and

00:05:22.971 --> 00:05:23.935
you will get the right answer if you

00:05:23.935 --> 00:05:24.690
do it correctly.

00:05:24.690 --> 00:05:26.896
But a simpler thing might be,

00:05:26.896 --> 00:05:29.833
well, let's just divide both sides by 0.5.

00:05:29.833 --> 00:05:30.727
That way I'm going to get more

00:05:30.727 --> 00:05:32.666
whole numbers involved.

00:05:32.666 --> 00:05:33.895
So if I divide.

00:05:33.895 --> 00:05:34.592
Remember, whatever I do to the

00:05:34.592 --> 00:05:35.451
left-hand side I have to do to the

00:05:35.451 --> 00:05:36.995
right-hand side.

00:05:36.995 --> 00:05:38.806
And the way my brain thought about it was,

00:05:38.806 --> 00:05:40.385
well, if I divide by 0.5 on the left-hand side

00:05:40.385 --> 00:05:41.895
I can get rid of this.

00:05:41.895 --> 00:05:43.844
And if I divide by 0.5 on the right-hand side

00:05:43.844 --> 00:05:45.609
I'm still going to get an integer.

00:05:45.609 --> 00:05:47.931
Three divided by 0.5 is six.

00:05:47.931 --> 00:05:49.324
It's the same thing as three divided by

00:05:49.324 --> 00:05:50.253
a half.

00:05:50.253 --> 00:05:52.610
How many halves fit into three?

00:05:52.610 --> 00:05:54.386
Six halves fit into three.

00:05:54.386 --> 00:05:56.011
So this going to be six right over here.

00:05:56.011 --> 00:05:56.812
So these cancel out.

00:05:56.812 --> 00:05:59.342
And then this is going to be equal to six.

00:05:59.342 --> 00:06:02.866
So the whole thing is simplified now

00:06:02.866 --> 00:06:07.576
to r + 2.75 is going to be equal to

00:06:07.576 --> 00:06:11.007
is equal to six.

00:06:11.007 --> 00:06:12.527
And now to just isolate the r on the

00:06:12.527 --> 00:06:14.222
left-hand side, I can substract

00:06:14.222 --> 00:06:17.066
the 2.75 from the left.

00:06:17.066 --> 00:06:18.970
But, like we've seen multiple times,

00:06:18.970 --> 00:06:21.521
I can't just do it from,

00:06:21.521 --> 00:06:24.692
I can't just do it from the left.

00:06:24.692 --> 00:06:26.628
I'm having, my brain is malfunctioning.

00:06:26.628 --> 00:06:27.545
2.75.

00:06:27.545 --> 00:06:28.741
I can't just do it from the left.

00:06:28.741 --> 00:06:30.378
I have to do it from the right as well.

00:06:30.378 --> 00:06:31.841
2.75.

00:06:31.841 --> 00:06:36.141
So this simplifies to r,

00:06:36.141 --> 00:06:38.149
this simplifies to r,

00:06:38.149 --> 00:06:40.761
Is equal to -- what's six minus 2.75?

00:06:40.761 --> 00:06:42.363
Well if you want to do it in your head,

00:06:42.363 --> 00:06:44.139
six minus two would be four,

00:06:44.139 --> 00:06:46.240
and then if you take 0.75 from that

00:06:46.240 --> 00:06:49.003
it would be 3.25.

00:06:49.003 --> 00:06:49.758
If you don't feel comfortable

00:06:49.758 --> 00:06:50.779
doing it in your head, we could just

00:06:50.779 --> 00:06:51.499
write it out.

00:06:51.499 --> 00:06:56.151
6.00 - 2.75,

00:06:56.151 --> 00:06:58.113
be careful to align the decimals

00:06:58.113 --> 00:07:00.215
and then I've got to do some re-grouping.

00:07:00.215 --> 00:07:02.200
Let's see, I have zero hundreds,

00:07:02.200 --> 00:07:03.570
trying to subtract five hundreds.

00:07:03.570 --> 00:07:04.812
That's not going to work out.

00:07:04.812 --> 00:07:06.170
So I try to re-group from here,

00:07:06.170 --> 00:07:06.936
but I have nothing here.

00:07:06.936 --> 00:07:08.213
So let me re-group from here.

00:07:08.213 --> 00:07:09.130
One one.

00:07:09.130 --> 00:07:10.454
Let me take away one of these ones.

00:07:10.454 --> 00:07:11.475
I'm going to have five ones.

00:07:11.475 --> 00:07:12.439
And then that's going to be equivalent,

00:07:12.439 --> 00:07:13.994
so I have five ones here,

00:07:13.994 --> 00:07:15.596
and that one one I took away is going to be

00:07:15.596 --> 00:07:17.268
ten tenths.

00:07:17.268 --> 00:07:18.928
And then I can take one of those

00:07:18.928 --> 00:07:20.310
tenths away.

00:07:20.310 --> 00:07:21.269
So I'm going to have nine tenths.

00:07:21.269 --> 00:07:23.149
And that's going to be ten hundredths.

00:07:23.149 --> 00:07:25.030
And now I could subtract.

00:07:25.030 --> 00:07:26.679
Ten hundreths minus five hundredths

00:07:26.679 --> 00:07:28.594
is five hundredths.

00:07:28.594 --> 00:07:31.612
Nine tenths minus seven tenths is

00:07:31.612 --> 00:07:32.797
two tenths.

00:07:32.797 --> 00:07:34.213
My decimal is going to be there.

00:07:34.213 --> 00:07:36.175
Five minus two is three.

00:07:36.175 --> 00:07:38.032
3.25.

00:07:38.032 --> 00:07:39.530
And you can verify that.

00:07:39.530 --> 00:07:44.115
3.25 + 2.75 is six, times 0.5 is indeed

00:07:44.115 --> 00:07:45.926
equal to three.

00:07:45.926 --> 00:07:47.215
So we feel, once again,

00:07:47.215 --> 99:59:59.999
really good about this.


Examples of simplifying expressions involving rational numbers

WEBVTT
Kind: captions
Language: en

00:00:00.247 --> 00:00:01.269
- [Voiceover] What I wanna do in this video

00:00:01.269 --> 00:00:03.267
is get some practice simplifying expressions

00:00:03.267 --> 00:00:04.948
and have some hairier numbers involved.

00:00:04.948 --> 00:00:07.189
These numbers are kind of hairy.

00:00:07.189 --> 00:00:09.047
Like always, try to pause this video

00:00:09.047 --> 00:00:10.719
and see if you can simplify this expression

00:00:10.719 --> 00:00:13.093
before I take a stab at it.

00:00:13.180 --> 00:00:15.502
All right, I'm assuming you have attempted it.

00:00:15.502 --> 00:00:16.558
Now let's look at it.

00:00:16.558 --> 00:00:23.745
We have -5.55 minus 8.55c plus 4.35c.

00:00:23.745 --> 00:00:25.748
So the first thing I wanna do is can I combine

00:00:25.748 --> 00:00:28.868
these c terms, and I definitely can.

00:00:30.148 --> 00:00:35.392
We can add -8.55c to 4.35c first,

00:00:35.392 --> 00:00:36.472
and then that would be, let's see,

00:00:36.472 --> 00:00:41.708
that would be -8.55 plus 4.35,

00:00:41.708 --> 00:00:44.250
I'm just adding the coefficients,

00:00:44.250 --> 00:00:46.386
times c, and of course, we still have

00:00:46.386 --> 00:00:48.894
that -5.55 out front.

00:00:48.894 --> 00:00:52.133
-5.55.

00:00:52.133 --> 00:00:54.164
I'll just put a plus there.

00:00:54.164 --> 00:00:56.405
Now how do we calculate -8.55

00:00:56.405 --> 00:00:58.727
plus 4.35?

00:00:58.727 --> 00:01:00.108
Well there's a couple of ways to think about it

00:01:00.108 --> 00:01:01.293
or visualize it.

00:01:01.293 --> 00:01:04.171
One way is to say well this is the same thing

00:01:04.171 --> 00:01:07.851
as the negative of 8.55

00:01:07.851 --> 00:01:10.951
minus 4.35,

00:01:10.951 --> 00:01:14.027
and 8.55 minus 4.35,

00:01:14.027 --> 00:01:15.989
let's see, eight minus four is going to be

00:01:15.989 --> 00:01:18.508
the negative, eight minus four is four,

00:01:18.508 --> 00:01:20.505
55 hundredths minus 35 hundredths

00:01:20.505 --> 00:01:21.805
is 20 hundredths.

00:01:21.805 --> 00:01:23.512
So I could write 4.20,

00:01:23.512 --> 00:01:26.310
which is really just the same thing as 4.2.

00:01:26.310 --> 00:01:28.840
So all of this can be replaced

00:01:28.840 --> 00:01:31.326
with a -4.2.

00:01:31.326 --> 00:01:33.147
So my entire expression has simplified

00:01:33.147 --> 00:01:39.301
to -5.55, and instead of saying plus -4.2c,

00:01:39.301 --> 00:01:46.289
I can just write it as minus 4.2c,

00:01:46.289 --> 00:01:46.985
and we're done.

00:01:46.985 --> 00:01:48.391
We can't simplify this anymore.

00:01:48.391 --> 00:01:50.410
We can't add this term that doesn't involve

00:01:50.410 --> 00:01:52.999
the variable to this term that does involve the variable.

00:01:52.999 --> 00:01:56.482
So this is about as simple as we're gonna get.

00:01:56.482 --> 00:01:58.671
So let's do another example.

00:01:58.671 --> 00:02:03.489
So here I have some more hairy numbers involved.

00:02:03.489 --> 00:02:05.532
These are all expressed as fractions.

00:02:05.532 --> 00:02:10.071
And so, let's see, I have 2/5m minus 4/5 minus 3/5m.

00:02:10.071 --> 00:02:11.615
So how can I simplify?

00:02:11.615 --> 00:02:14.273
Well I could add all the m terms together.

00:02:14.273 --> 00:02:15.550
So let me just change the order.

00:02:15.550 --> 00:02:20.949
I could rewrite this as 2/5m minus 3/5m

00:02:20.949 --> 00:02:22.237
minus 4/5.

00:02:22.237 --> 00:02:24.373
All I did was I changed the order.

00:02:24.373 --> 00:02:26.846
We can see that I have these two m terms.

00:02:26.846 --> 00:02:28.773
I can add those two together.

00:02:28.773 --> 00:02:33.626
So this is going to be 2/5 minus 3/5

00:02:33.626 --> 00:02:39.453
times m, and then I have the -4/5

00:02:39.453 --> 00:02:41.311
still on the right hand side.

00:02:41.311 --> 00:02:44.352
Now what's 2/5 minus 3/5?

00:02:44.352 --> 00:02:46.930
Well that's gonna be -1/5.

00:02:46.930 --> 00:02:48.752
That's gonna be -1/5.

00:02:48.752 --> 00:02:54.673
So I have -1/5m minus 4/5.

00:02:54.673 --> 00:02:57.088
Minus 4/5.

00:02:57.088 --> 00:02:57.831
Now once again, I'm done.

00:02:57.831 --> 00:02:59.003
I can't simplify it anymore.

00:02:59.003 --> 00:03:01.824
I can't add this term that involves m somehow

00:03:01.824 --> 00:03:04.088
to this -4/5.

00:03:04.088 --> 00:03:06.850
So we are done here.

00:03:06.850 --> 00:03:09.091
Let's do one more.

00:03:09.091 --> 00:03:11.309
Let's do one more example.

00:03:11.309 --> 00:03:12.992
So here, and this is interesting,

00:03:12.992 --> 00:03:16.311
I have a parentheses and all the rest.

00:03:16.311 --> 00:03:17.647
Like always, pause the video.

00:03:17.647 --> 00:03:20.103
See if you can simplify this.

00:03:20.103 --> 00:03:22.402
All right, let's work through it together.

00:03:22.402 --> 00:03:23.702
Now the first thing that I want to do

00:03:23.702 --> 00:03:25.606
is let's distribute this two so that we

00:03:25.606 --> 00:03:27.405
just have three terms that are just

00:03:27.405 --> 00:03:29.135
being added and subtracted.

00:03:29.135 --> 00:03:31.098
So if we distribute this two,

00:03:31.098 --> 00:03:33.433
we're gonna get two times 1/5m

00:03:33.433 --> 00:03:36.312
is 2/5m.

00:03:36.312 --> 00:03:37.834
Let me make sure you see that m.

00:03:37.834 --> 00:03:38.763
M is right here.

00:03:38.763 --> 00:03:43.220
Two times -2/5 is -4/5,

00:03:43.220 --> 00:03:45.542
and then I have plus 3/5.

00:03:45.542 --> 00:03:47.597
Now how can we simplify this more?

00:03:47.597 --> 00:03:48.793
Well I have these two terms here

00:03:48.793 --> 00:03:50.163
that don't involve the variable.

00:03:50.163 --> 00:03:51.498
Those are just numbers.

00:03:51.498 --> 00:03:53.297
I can add them to each other.

00:03:53.297 --> 00:03:56.478
So I have -4/5 plus 3/5.

00:03:56.478 --> 00:03:58.811
So what's negative four plus three?

00:03:58.811 --> 00:04:00.077
That's going to be negative one.

00:04:00.077 --> 00:04:02.573
So this is going to be -1/5,

00:04:02.573 --> 00:04:04.093
what we have in yellow here.

00:04:04.093 --> 00:04:08.656
I still have the 2/5m,

00:04:08.656 --> 00:04:10.838
2/5m minus 1/5.

00:04:10.838 --> 00:04:11.616
And we're done.

00:04:11.616 --> 00:04:14.630
We've simplified that as much as we can.


Equivalent expressions with distribution and negative numbers

WEBVTT
Kind: captions
Language: en

00:00:00.260 --> 00:00:01.409
- Let's get some practice

00:00:01.409 --> 00:00:03.603
identifying equivalent expressions.

00:00:03.603 --> 00:00:06.180
So I have an expression written here in yellow

00:00:06.180 --> 00:00:08.549
and then I have two more written in this light green color.

00:00:08.549 --> 00:00:10.652
And I want you to pause this video and see if you can

00:00:10.652 --> 00:00:12.870
figure out which of these expressions,

00:00:12.870 --> 00:00:14.715
and it's possible that neither of them are,

00:00:14.715 --> 00:00:18.326
which of these are equivalent to the one in yellow.

00:00:19.082 --> 00:00:20.649
So I'm assuming you've had a go at it.

00:00:20.649 --> 00:00:22.561
So the way I like to tackle it is just

00:00:22.561 --> 00:00:24.639
to simplify all of them as much as possible.

00:00:24.639 --> 00:00:27.324
So this one up here is clearly not that simplified.

00:00:27.324 --> 00:00:29.551
So let's distribute this two.

00:00:29.551 --> 00:00:32.123
So if I distribute the two, what does it become?

00:00:32.123 --> 00:00:36.703
This is equal to two times negative six-C, is negative 12-C.

00:00:36.703 --> 00:00:40.609
Two times positive three is positive six.

00:00:40.609 --> 00:00:44.100
And then we have plus four-C, plus four-C.

00:00:44.775 --> 00:00:46.689
And then we can simplify it further

00:00:46.689 --> 00:00:49.138
cause I have both of these terms that involve C.

00:00:49.138 --> 00:00:52.963
I have negative 12-C plus four-C.

00:00:52.963 --> 00:00:54.402
So what's that going to be?

00:00:54.402 --> 00:00:57.692
Negative 12 of something plus four of something

00:00:57.692 --> 00:00:59.352
is going to be negative eight of that something.

00:00:59.352 --> 00:01:01.880
So this is going to be equal to negative eight-C.

00:01:01.880 --> 00:01:04.306
So these two blue terms when I add them,

00:01:04.306 --> 00:01:05.804
I'm going to get negative eight-C.

00:01:05.804 --> 00:01:07.584
And then finally plus six.

00:01:08.296 --> 00:01:09.501
Plus six.

00:01:10.580 --> 00:01:12.637
Now just doing that that's exactly

00:01:12.637 --> 00:01:14.518
what this first green expression is.

00:01:14.518 --> 00:01:16.888
So this one is definitely, this one is

00:01:16.888 --> 00:01:18.583
definitely going to be equivalent.

00:01:18.583 --> 00:01:20.662
Now what about this one down here?

00:01:20.662 --> 00:01:22.848
Well to figure that out let's simplify it.

00:01:22.848 --> 00:01:24.536
So let's distribute the three.

00:01:24.536 --> 00:01:27.751
Three times negative four-C is negative 12-C.

00:01:27.751 --> 00:01:31.593
Three times positive two is positive six.

00:01:31.593 --> 00:01:35.848
So plus six and then we have the plus four-C over there.

00:01:35.848 --> 00:01:37.521
It's lookin' good.

00:01:37.521 --> 00:01:41.164
And then we can add the terms that involve C.

00:01:41.164 --> 00:01:44.995
Negative 12-C and four-C, you add those together,

00:01:44.995 --> 00:01:47.231
you're gonna get negative eight-C.

00:01:47.231 --> 00:01:49.283
Negative eight-C plus six.

00:01:50.353 --> 00:01:54.454
Plus six, which is exactly what these other ones are.

00:01:54.454 --> 00:01:57.864
So all of these, all of these expressions,

00:01:57.864 --> 00:01:59.584
are actually equivalent.

00:01:59.584 --> 00:02:01.686
This one, that one, and that one.

00:02:01.686 --> 00:02:03.414
Let's do another example.

00:02:04.259 --> 00:02:06.403
And just like the last time pause the video

00:02:06.403 --> 00:02:08.737
and see which of these two expressions,

00:02:08.737 --> 00:02:10.490
it could be both of them or it could be none of them,

00:02:10.490 --> 00:02:11.583
or it could be one of them.

00:02:11.583 --> 00:02:13.871
Which of them, if any, are equivalent

00:02:13.871 --> 00:02:15.668
to this yellow expression?

00:02:16.316 --> 00:02:17.924
Alright let's do it together.

00:02:17.924 --> 00:02:19.778
And like before let's just simplify it.

00:02:19.778 --> 00:02:22.582
So the first thing my brain wants to do

00:02:22.582 --> 00:02:26.593
is let's take the terms involving N and add those together.

00:02:26.593 --> 00:02:29.962
So negative six of something, in this case N,

00:02:29.962 --> 00:02:31.749
plus four of that something, in that case N.

00:02:31.749 --> 00:02:34.177
So negative six-N plus four-N that's gonna leave you

00:02:34.177 --> 00:02:35.687
with negative two of that something.

00:02:35.687 --> 00:02:36.586
You add the coefficients.

00:02:36.586 --> 00:02:39.922
Negative six plus four is negative two Ns.

00:02:39.922 --> 00:02:43.054
So we have negative two-N, and then plus negative 12,

00:02:43.054 --> 00:02:45.235
that's the same thing as just minus 12.

00:02:45.235 --> 00:02:48.123
So minus, minus 12.

00:02:48.123 --> 00:02:50.913
So I simplified our original expression.

00:02:50.913 --> 00:02:53.600
Let's see these ones, these down here.

00:02:53.600 --> 00:02:57.279
So if I distribute the four, if I distribute the four

00:02:57.279 --> 00:03:00.181
I get four times N is four-N.

00:03:00.181 --> 00:03:03.373
And then four times negative three is minus 12.

00:03:03.373 --> 00:03:06.047
And then we are going to subtract six-N.

00:03:06.804 --> 00:03:09.576
So minus six-N.

00:03:09.576 --> 00:03:10.667
So what does this give us?

00:03:10.667 --> 00:03:13.354
We get, let me get another color here.

00:03:13.354 --> 00:03:16.599
So we have four-N, I'm adding all the terms with N,

00:03:16.599 --> 00:03:20.499
minus six-N, that's gonna give us negative two-N.

00:03:20.499 --> 00:03:22.624
And then we have the minus 12.

00:03:22.624 --> 00:03:25.115
And then we have the minus 12.

00:03:25.115 --> 00:03:27.359
So this expression when I simplify it got me

00:03:27.359 --> 00:03:29.170
the exact same place as the first expression.

00:03:29.170 --> 00:03:32.034
So these two, these two are equivalent.

00:03:32.034 --> 00:03:33.485
This is equivalent to that.

00:03:33.485 --> 00:03:35.196
Now let's check this one out.

00:03:35.196 --> 00:03:37.603
So two, let me just distribute,

00:03:37.603 --> 00:03:39.901
let me just distribute the two.

00:03:39.901 --> 00:03:41.944
Two times two-N is four-N.

00:03:41.944 --> 00:03:45.576
And then two times negative six is negative 12.

00:03:45.576 --> 00:03:49.440
So this simplified to four-N minus 12 which is clearly

00:03:49.440 --> 00:03:52.261
different than negative two-N minus 12.

00:03:52.261 --> 00:03:54.359
So this one, this one,

00:03:54.359 --> 00:03:58.006
is not the same as the other two.


Mileage assumption example

WEBVTT
Kind: captions
Language: en

00:00:00.750 --> 00:00:03.260
Mary has decided
to buy a new car.

00:00:03.260 --> 00:00:06.670
The local car dealership states
the car she is considering

00:00:06.670 --> 00:00:10.860
has an estimated gas mileage
of 24 miles per gallon.

00:00:10.860 --> 00:00:13.620
The car has a 12 gallon tank.

00:00:13.620 --> 00:00:15.380
Mary tells her
friend that the car

00:00:15.380 --> 00:00:20.110
will travel exactly 288
miles on a full tank of gas.

00:00:20.110 --> 00:00:21.610
Is this a reasonable statement?

00:00:21.610 --> 00:00:22.984
Why or why not?

00:00:22.984 --> 00:00:25.150
And I encourage you to pause
this video now and look

00:00:25.150 --> 00:00:26.250
at these statements.

00:00:26.250 --> 00:00:28.310
And think about is this
a reasonable statement,

00:00:28.310 --> 00:00:29.620
yes or no?

00:00:29.620 --> 00:00:30.960
And then justify.

00:00:30.960 --> 00:00:32.560
Use one of these
justifications to say

00:00:32.560 --> 00:00:35.890
whether it is a reasonable
statement or not.

00:00:35.890 --> 00:00:38.047
So let's read
through each of them.

00:00:38.047 --> 00:00:40.380
So the first one says, yes,
it is a reasonable statement

00:00:40.380 --> 00:00:44.510
for Mary to tell her friend that
the car will travel exactly 288

00:00:44.510 --> 00:00:46.320
miles on a full tank of gas.

00:00:46.320 --> 00:00:49.180
And the rationale is, the
car can travel exactly 24

00:00:49.180 --> 00:00:51.670
miles per gallon
times 12 gallons.

00:00:51.670 --> 00:00:53.970
We see that the units, the
gallon in the numerator,

00:00:53.970 --> 00:00:55.540
gallon in the
denominator cancel out.

00:00:55.540 --> 00:00:59.480
And we're left with 24
times 12, which is 288.

00:00:59.480 --> 00:01:01.730
And the units we're
left with is miles.

00:01:01.730 --> 00:01:06.370
288 miles on 1 tank of gas.

00:01:06.370 --> 00:01:08.340
Now there's one
little thing that

00:01:08.340 --> 00:01:10.360
makes this-- the
math seems right.

00:01:10.360 --> 00:01:12.030
The unit cancellation
seems right,

00:01:12.030 --> 00:01:14.790
the dimensional
analysis, so to speak.

00:01:14.790 --> 00:01:19.500
But it seems like the statement
is using a stronger assumption

00:01:19.500 --> 00:01:21.540
than what the car
dealer told her.

00:01:21.540 --> 00:01:25.110
The car dealer told her
that it has an estimated gas

00:01:25.110 --> 00:01:27.055
mileage of 24 miles per gallon.

00:01:29.620 --> 00:01:31.290
To do this, she
would have to assume

00:01:31.290 --> 00:01:34.560
that the car can travel
exactly 24 miles in a gallon.

00:01:34.560 --> 00:01:36.410
There's a big difference
between an estimate

00:01:36.410 --> 00:01:40.080
and being exactly
24 miles per gallon.

00:01:40.080 --> 00:01:42.860
In the estimate, first of all,
they might have rounded it.

00:01:42.860 --> 00:01:45.720
When they did the tests,
it might have been 23.9.

00:01:45.720 --> 00:01:49.520
It might have even have
been 23.6 miles per gallon.

00:01:49.520 --> 00:01:51.739
Or it might have been
24.4 miles per gallon.

00:01:51.739 --> 00:01:53.030
That's a pretty big difference.

00:01:53.030 --> 00:01:56.610
You definitely wouldn't
get to exactly 288 miles.

00:01:56.610 --> 00:01:59.000
The other thing is this
estimate is probably

00:01:59.000 --> 00:02:01.961
based on an average of
traveling many, many, many miles

00:02:01.961 --> 00:02:02.960
in different conditions.

00:02:02.960 --> 00:02:05.790
You get better gas mileage
if you're on the highway,

00:02:05.790 --> 00:02:07.310
or if you're going downhill.

00:02:07.310 --> 00:02:09.050
And you get worse
gas mileage if you're

00:02:09.050 --> 00:02:10.770
in traffic, or if
you're going uphill,

00:02:10.770 --> 00:02:12.790
or if there's a strong headwind.

00:02:12.790 --> 00:02:14.500
So it just averages here.

00:02:14.500 --> 00:02:17.590
So you can't say that it's
going to go exactly 288

00:02:17.590 --> 00:02:19.969
miles without knowing exactly
how this was measured.

00:02:19.969 --> 00:02:22.510
And frankly, every car is going
to be a little bit different.

00:02:22.510 --> 00:02:25.150
The octane of the fuel will
be a little bit different.

00:02:25.150 --> 00:02:28.830
And so that's why it's hard to
make this assumption right over

00:02:28.830 --> 00:02:32.270
here, that the car can travel
exactly 24 miles per gallon.

00:02:32.270 --> 00:02:34.940
The dealership just
gave an estimate.

00:02:34.940 --> 00:02:37.153
So let's go to B. Yes,
the estimated gas mileage

00:02:37.153 --> 00:02:40.010
of the car can be used to
calculate the exact distance

00:02:40.010 --> 00:02:42.220
the car will travel on
any one tank of gas.

00:02:42.220 --> 00:02:44.690
Well, no, we just already
said it's an estimate.

00:02:44.690 --> 00:02:48.200
It's roughly how far the car
can travel on a tank of gas,

00:02:48.200 --> 00:02:50.200
but it's not the exact
distance the car will

00:02:50.200 --> 00:02:52.940
travel on any one tank of gas.

00:02:52.940 --> 00:02:54.820
C, no, it's not a
reasonable statement.

00:02:54.820 --> 00:02:57.500
24 miles per gallon is an
estimate, likely rounded

00:02:57.500 --> 00:02:59.166
to the nearest integer,
so it would not

00:02:59.166 --> 00:03:00.540
be reasonable to
try to calculate

00:03:00.540 --> 00:03:02.441
the exact distance
the car can travel.

00:03:02.441 --> 00:03:03.940
That's what I think
I've been saying

00:03:03.940 --> 00:03:05.540
all along throughout this video.

00:03:05.540 --> 00:03:07.350
So let's just read
D. No, the distance

00:03:07.350 --> 00:03:10.460
the car will travel is found
by dividing 24 miles per gallon

00:03:10.460 --> 00:03:11.280
by 12 gallons.

00:03:11.280 --> 00:03:12.560
No, this is just wrong.

00:03:12.560 --> 00:03:15.500
This is wrong
mathematically, and this

00:03:15.500 --> 00:03:18.810
is just troublesome
on a lot of levels.

00:03:18.810 --> 00:03:19.950
C is the right answer.

00:03:19.950 --> 00:03:25.924
D is the most wrong of
all the answers here.


Finding the best deal on pesticides

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:04.480
A farm supply store carries four
different brands of pesticide.

00:00:04.480 --> 00:00:08.210
Each brand sells pesticide
of a different concentration.

00:00:08.210 --> 00:00:10.900
And each brand also has a
different size of bottle.

00:00:10.900 --> 00:00:12.640
So here we have the
different brands,

00:00:12.640 --> 00:00:15.030
they give us the bottle
size, the concentration,

00:00:15.030 --> 00:00:20.220
so that's the percentage of this
fluid volume that is actually

00:00:20.220 --> 00:00:22.680
pesticide that's actually
active ingredient.

00:00:22.680 --> 00:00:25.302
And then they tell us
the price per bottle.

00:00:25.302 --> 00:00:27.510
And they ask us if we want
the most active ingredient

00:00:27.510 --> 00:00:29.260
for our money, you
should spend your money

00:00:29.260 --> 00:00:33.180
on bottles of brand
question mark.

00:00:33.180 --> 00:00:36.330
So let's just think about
which of these brands

00:00:36.330 --> 00:00:39.930
gives us the most active
ingredient for our money.

00:00:39.930 --> 00:00:41.960
And I encourage you
to pause this video

00:00:41.960 --> 00:00:44.740
and try to think about
this on your own.

00:00:44.740 --> 00:00:58.990
Well, let's just think
about active ingredient.

00:00:58.990 --> 00:01:01.250
I'll just write
ingred for short.

00:01:01.250 --> 00:01:05.610
Ingredient per bottle.

00:01:05.610 --> 00:01:08.730
And I'll do this in milliliters.

00:01:08.730 --> 00:01:13.130
So, we have 375 milliliters, 65%
of which is going to be active.

00:01:13.130 --> 00:01:30.230
So 65% of 375 gets
us 243.75 milliliters

00:01:30.230 --> 00:01:32.280
of active ingredient per bottle.

00:01:32.280 --> 00:01:42.395
How much is that active
ingredient per dollar here?

00:01:42.395 --> 00:01:44.020
Well, you could just
reason through it.

00:01:44.020 --> 00:01:49.480
You say I have
273.75 mLs for $22.

00:01:49.480 --> 00:01:51.950
So if I want per dollar,
I'll just divide this by 22.

00:01:54.460 --> 00:01:55.080
So let me see.

00:01:55.080 --> 00:01:56.970
If I just do that as
my previous answer,

00:01:56.970 --> 00:02:00.790
which is that divided
by 22, and I'll just

00:02:00.790 --> 00:02:05.620
round to the nearest
penny, $11.08.

00:02:05.620 --> 00:02:06.120
Let me see.

00:02:06.120 --> 00:02:06.920
This is in mLs.

00:02:06.920 --> 00:02:09.710
11.07.

00:02:09.710 --> 00:02:12.420
Once again, this is
in mLs, not dollars.

00:02:12.420 --> 00:02:15.170
This is mLs per bottle,
milliliters per bottle,

00:02:15.170 --> 00:02:21.092
divided by $22 per bottle,
gets us 11.07 mLs per dollar.

00:02:21.092 --> 00:02:22.550
Actually, let me
write it that way.

00:02:22.550 --> 00:02:26.710
mLs per dollar, just so we
can keep track of the unit.

00:02:26.710 --> 00:02:30.050
This is mLs per bottle.

00:02:30.050 --> 00:02:32.630
Now another way that we could
have thought about it is look.

00:02:32.630 --> 00:02:40.220
You have 273.75 mLs per bottle.

00:02:40.220 --> 00:02:45.740
We want to get it
to mLs per dollar.

00:02:45.740 --> 00:02:48.500
So we want to get rid of a
bottle in the denominator.

00:02:48.500 --> 00:02:50.470
So we would want to
multiply it by something

00:02:50.470 --> 00:02:52.210
with bottles in the numerator.

00:02:52.210 --> 00:02:54.530
And we want to introduce a
dollar in the denominator,

00:02:54.530 --> 00:02:57.480
so it would be
bottles per dollar.

00:02:57.480 --> 00:02:59.250
And how many bottles
are there per dollar?

00:02:59.250 --> 00:03:03.690
Or another way to think about
it is one bottle is $22.

00:03:03.690 --> 00:03:10.530
So you multiply 23.75 times 1
over 22, you would get 11.07.

00:03:10.530 --> 00:03:14.634
And that cancels with that,
milliliters per dollar.

00:03:14.634 --> 00:03:17.050
And notice, you could have
done this through common sense.

00:03:17.050 --> 00:03:18.680
This is equivalent
to just dividing.

00:03:18.680 --> 00:03:25.740
This is how much you get for
a $22 bottle, divided by $22,

00:03:25.740 --> 00:03:29.030
you get the
milliliters per dollar.

00:03:29.030 --> 00:03:31.210
But either way, let's do
it for the rest of them.

00:03:31.210 --> 00:03:35.200
So for this one right over here,
in general, to figure this out,

00:03:35.200 --> 00:03:37.710
we just have to multiply the
concentration times the bottle

00:03:37.710 --> 00:03:38.210
size.

00:03:38.210 --> 00:03:39.550
And then divide it by the price.

00:03:39.550 --> 00:03:41.460
So let's just do that
for the next three.

00:03:41.460 --> 00:03:43.900
So the concentration is 35%.

00:03:47.330 --> 00:03:52.770
So 0.35 times 750.

00:03:52.770 --> 00:03:54.830
So this expression
is going to give us

00:03:54.830 --> 00:03:57.020
the milliliters of
active ingredient.

00:03:57.020 --> 00:04:01.200
And then we can divide
by the cost per bottle.

00:04:01.200 --> 00:04:04.400
So this is 12.5
milliliters per dollar.

00:04:11.190 --> 00:04:13.490
Now this next one,
they give us one liter.

00:04:13.490 --> 00:04:17.380
But we know one liter is
equal to 1,000 milliliters.

00:04:17.380 --> 00:04:20.000
75% of that is going to
be active ingredient,

00:04:20.000 --> 00:04:22.570
so we're going to
have 750 milliliters

00:04:22.570 --> 00:04:25.026
of active ingredient.

00:04:25.026 --> 00:04:26.150
And it's going to cost $68.

00:04:26.150 --> 00:04:28.520
So we just divide this by 68.

00:04:28.520 --> 00:04:35.760
And we get 750
milliliters divided by 68,

00:04:35.760 --> 00:04:41.040
it gets us $11, and
if we round, $0.3.

00:04:41.040 --> 00:04:42.220
We're in milliliters.

00:04:42.220 --> 00:04:48.670
11.03 milliliters per dollar.

00:04:48.670 --> 00:04:51.070
Have to be very careful.

00:04:51.070 --> 00:04:52.390
This is milliliters per dollar.

00:04:52.390 --> 00:04:55.020
That's the whole point of this
is to keep track of the units.

00:04:55.020 --> 00:04:58.510
So this isn't dollars, this
is milliliters per dollar.

00:04:58.510 --> 00:05:00.760
And then finally, 1.5 liters.

00:05:00.760 --> 00:05:07.160
We'll times 1,000
milliliters per liter.

00:05:07.160 --> 00:05:11.360
This would give us
1,500 milliliters,

00:05:11.360 --> 00:05:13.860
30% of which is active.

00:05:13.860 --> 00:05:16.410
So 30% of 1,500 milliliters.

00:05:16.410 --> 00:05:19.690
That's 450 milliliters.

00:05:19.690 --> 00:05:23.110
And then we can
divide that by $40.

00:05:23.110 --> 00:05:38.760
So 450 divided by $40 gets us
to 11.25 milliliters per dollar.

00:05:38.760 --> 00:05:40.156
So which one is the best deal?

00:05:40.156 --> 00:05:41.656
Well, the best deal
is the one where

00:05:41.656 --> 00:05:44.200
I get the most
milliliters per dollar.

00:05:44.200 --> 00:05:45.970
And that's brand
b right over here.

00:05:45.970 --> 00:05:49.860
I get 12.5 milliliters
per dollar.

00:05:49.860 --> 00:05:53.603
I'm getting the most
pesticide for my buck.