Equations of parallel and perpendicular lines

WEBVTT
Kind: captions
Language: en

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In this video we're going to do
a couple of examples that

00:00:02.990 --> 00:00:06.180
deal with parallel and
perpendicular lines.

00:00:06.180 --> 00:00:12.960
So you have parallel, you have
perpendicular, and, of course,

00:00:12.960 --> 00:00:16.129
you have lines that are
neither parallel nor

00:00:16.129 --> 00:00:16.950
perpendicular.

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And just as a bit of review,
if you've never seen this

00:00:19.120 --> 00:00:22.510
before, parallel lines,
they never intersect.

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So let me draw some axes.

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So if those are my coordinate
axes right there, that's my

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x-axis, that is my y-axis.

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If this is a line that I'm
drawing in magenta, a parallel

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line might look something
like this.

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It's not the exact same
line, but they have

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the exact same slope.

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If this moves a certain amount,
if this change in y

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over change in x is a certain
amount, this change in y over

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change in x is the
same amount.

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And that's why they
never intersect.

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So they have the same slope.

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Parallel lines have
the same slope.

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Perpendicular lines, depending
on how you want to view it,

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they're kind of the opposite.

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Let's say that this
is some line.

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A line that is perpendicular
to that will not only

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intersect the line, it will
intersect it at a right angle,

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at a 90 degree angle.

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And I'm not going to prove
it for you here.

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I actually prove it in the
linear algebra playlist. But a

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perpendicular line's slope--
so let's say that this one

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right here, let's say that
yellow line has a slope of m.

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Then this orange line, that's
perpendicular to the yellow

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line, is going to have a slope
of negative 1 over m.

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Their slopes are going
to be the negative

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inverse of each other.

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Now, given this information,
let's look at a bunch of lines

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and figure out if they're
parallel, if they're

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perpendicular, or if
they are neither.

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And to do that, we just have to
keep looking at the slopes.

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So let's see, they say one
line passes through the

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points, 4, negative 3,
and negative 8, 0.

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Another line passes through
the points, negative 1,

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negative 1, and negative 2, 6.

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So let's figure out the slopes
of each of these lines.

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So I'll first do this
one in pink.

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So this slope right here,
so line 1, so I'll

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call it slope 1.

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Slope 1 is, let's just say it
is-- well, I'll take this as

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the finishing point.

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So negative 3 minus 0--
remember change in y--

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negative 3 minus 0, over
4 minus negative 8.

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So this is equal to negative 3
over-- this is the same thing

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as 4 plus 8-- negative
3 over 12, which is

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equal to negative 1/4.

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Divide the numerator and
the denominator by 3.

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That's this line.

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That's the first line.

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Now, what about the
second line?

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The slope for that second line
is, well, let's take, here,

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negative 1 minus 6, over
negative 1 minus negative 2 is

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equal to-- negative 1 minus 6 is
negative 7, over negative 1

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minus negative 2.

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That's the same thing as
negative 1 plus 2.

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Well, that's just 1.

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So the slope here
is negative 7.

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So here, their slopes are
neither equal-- so they're not

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parallel-- nor are
they the negative

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inverse of each other.

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So this is neither.

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This is neither parallel
nor perpendicular.

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So these two lines, they
intersect, but they're not

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going to intersect at
a 90 degree angle.

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Let's do a couple
more of these.

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So I have here, once again, one
line passing through these

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points, and then another line
passing through these points.

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So let's just look
at their slopes.

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So this one in the green,
what's the slope?

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The slope of the green one, I'll
call that the first line.

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We could say, let's
see, change in y.

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So we could do negative 2
minus 14, over-- I did

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negative 2 first, so I'll
do 1 first-- over 1

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minus negative 3.

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So negative 2 minus
14 is negative 16.

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1 minus negative 3 is the
same things as 1 plus 3.

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That's over 4.

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So this is negative 4.

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Now, what's the slope of that
second line right there?

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So we have the slope of
that second line.

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Let's say 5 minus negative 3,
that's our change in y, over

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negative 2 minus 0.

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So this is equal to 5
minus negative 3.

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That's the same thing
as 5 plus 3.

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That's 8.

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And then negative 2 minus
0 is negative 2.

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So this is also equal
to negative 4.

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So these two lines
are parallel.

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They have the exact
same slope.

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And I encourage you to find the
equations of both of these

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lines and graph both of these
lines, and verify for yourself

00:05:28.790 --> 00:05:32.060
that they are indeed parallel.

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Let's do this one.

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Once again, this is just an
exercise in finding slope.

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So this first line
has those points.

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Let's figure out its slope.

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The slope of this first
line, one line passes

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through these points.

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So let's see, 3 minus negative
3, that's our change in y,

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over 3 minus negative 6.

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So this is the same thing as 3
plus 3, which is 6, over 3

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plus 6, which is 9.

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So this first line has
a slope of 2/3.

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What is the second
line's slope?

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This was the second line there,
that's the other line

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passing through these points.

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So the other line's slope,
let's see, we could say

00:06:15.550 --> 00:06:27.340
negative 8 minus 4, over
2 minus negative 6.

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So what is this equal to?

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Negative 8 minus 4
is negative 12.

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2 minus negative 6, that's the
same thing as 2 plus 6.

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The negatives cancel out.

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So it's negative 12 over 8,
which is the same thing if we

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divide the numerator and the
denominator by 4, that's

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negative 3/2.

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Notice, these guys
are the negative

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inverse of each other.

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If I take negative 1 over 2/3,
that is equal to negative 1

00:06:57.850 --> 00:07:02.870
times 3/2, which is equal
to negative 3/2.

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These guys are the negative
inverses of each other.

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You swap the numerator and the
denominator, make them

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negative, and they become
equal to each other.

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So these two lines are
perpendicular.

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And I encourage you to find the
equations-- I already got

00:07:18.150 --> 00:07:20.560
the slopes for you-- but find
the equations of both of these

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lines, plot them, and verify
for yourself that they are

00:07:24.090 --> 00:07:25.620
perpendicular.

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Let's do one more.

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Find the equation of a line
perpendicular to this line

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that passes through the
point 2 comma 8.

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So this first piece of
information, that it's

00:07:37.440 --> 00:07:40.650
perpendicular to that line right
over there, what does

00:07:40.650 --> 00:07:41.810
that tell us?

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Well, if it's perpendicular to
this line, its slope has to be

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the negative inverse of 2/5.

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So its slope, the negative
inverse of 2/5, the inverse of

00:07:49.920 --> 00:07:54.510
2/5 is-- let me do it in a
better color, a nicer green.

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If this line's slope is negative
2/5, the equation of

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the line we have to figure out
that's perpendicular, its

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slope is going to
be the inverse.

00:08:01.970 --> 00:08:05.180
So instead of 2/5, it's
going to be 5/2.

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Instead of being a negative,
it's going to be a positive.

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So this is a negative inverse
of negative 2/5.

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Right?

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You take the negative sign
and it becomes positive.

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You swap the 5 and the
2, you get 5/2.

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So that is going to have
to be our slope.

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And we can actually use the
point-slope form right here.

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It goes through this
point right there.

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So let's use point-slope form.

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y minus this y-value, which
has to be on the line, is

00:08:32.220 --> 00:08:38.880
equal to our slope, 5/2 times
x minus this x-value, the

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x-value when y is equal to 8.

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And this is the equation of the
line in point-slope form.

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If you want to put it in
slope-intercept form, you can

00:08:46.940 --> 00:08:49.960
just do a little bit of
algebra, algebraic

00:08:49.960 --> 00:08:52.910
manipulation. y minus 8 is equal
to-- let's distribute

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the 5/2-- so 5/2 x minus--
5/2 times 2 is just-- 5.

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And then add 8 to both sides.

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You get y is equal to 5/2x.

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Add 8 to negative
5, so plus 3.

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And we are done.


Linear Equations in Standard Form

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:04.610
Let's do some examples dealing
with equations of lines in

00:00:04.610 --> 00:00:06.080
standard form.

00:00:06.080 --> 00:00:08.980
So, so far we've had two
other forms. We've had

00:00:08.980 --> 00:00:15.470
slope-intercept, which
is of the form, y is

00:00:15.470 --> 00:00:18.630
equal to mx plus b.

00:00:18.630 --> 00:00:19.920
That's actually this
right here.

00:00:19.920 --> 00:00:22.020
This is in slope-intercept
form.

00:00:22.020 --> 00:00:28.540
We've seen point-slope form
in the last video.

00:00:28.540 --> 00:00:34.190
That's of the form, y minus some
y-value on the line being

00:00:34.190 --> 00:00:37.960
equal to the slope times x
minus some x-value on the

00:00:37.960 --> 00:00:40.670
line, when you have
that y-value.

00:00:40.670 --> 00:00:43.830
So the point x1, y1
is on the line.

00:00:43.830 --> 00:00:46.820
This right here is an example
of point-slope form.

00:00:46.820 --> 00:00:49.180
And now we're going to talk
about the standard form.

00:00:49.180 --> 00:00:54.790
And the standard form-- let me
write it here-- standard form

00:00:54.790 --> 00:00:57.680
is essentially putting all of
the x and y terms onto the

00:00:57.680 --> 00:00:59.170
left-hand side of
the equation.

00:00:59.170 --> 00:01:05.570
So you get ax plus
by is equal to c.

00:01:05.570 --> 00:01:10.510
I want to really emphasize that
all of these are just

00:01:10.510 --> 00:01:14.190
different ways of writing
the same equation.

00:01:14.190 --> 00:01:16.655
If you're given this, you can
out algebraically manipulate

00:01:16.655 --> 00:01:18.510
it to get to that or to that.

00:01:18.510 --> 00:01:21.080
If you're given that, you
can get to that or that.

00:01:21.080 --> 00:01:24.590
These are all different ways
of writing the exact same

00:01:24.590 --> 00:01:27.110
relationship, the
exact same line.

00:01:27.110 --> 00:01:29.170
So let's do a couple of
examples of this.

00:01:29.170 --> 00:01:31.380
So here we have a
line right here.

00:01:31.380 --> 00:01:34.290
We have an equation written
in slope-intercept form.

00:01:34.290 --> 00:01:37.250
The slope is 3, the y-intercept
is negative 8.

00:01:37.250 --> 00:01:39.340
Let's put it into
standard form.

00:01:39.340 --> 00:01:41.270
So we just have to get
the 3x onto the

00:01:41.270 --> 00:01:42.690
other side of the equation.

00:01:42.690 --> 00:01:45.465
And the best way I can think of
doing that-- let me rewrite

00:01:45.465 --> 00:01:51.560
the equation, y is equal to 3x
minus 8-- let's some subtract

00:01:51.560 --> 00:01:54.180
3x from both sides
of the equation.

00:01:54.180 --> 00:01:58.500
So if you subtract 3x from both
sides-- so you subtract

00:01:58.500 --> 00:02:04.380
3x, subtract 3x-- what do the
left- and right-hand sides of

00:02:04.380 --> 00:02:06.600
the equation become?

00:02:06.600 --> 00:02:14.470
The left-hand side becomes
negative 3x plus y being equal

00:02:14.470 --> 00:02:18.560
to-- the 3x and the negative 3x
cancel out-- being equal to

00:02:18.560 --> 00:02:20.150
negative 8.

00:02:20.150 --> 00:02:21.050
We're done.

00:02:21.050 --> 00:02:24.590
That's standard form
right there.

00:02:24.590 --> 00:02:27.610
Standard form, I guess people
like it because it has both

00:02:27.610 --> 00:02:29.680
the coefficients on the
left-hand side.

00:02:29.680 --> 00:02:32.280
But it's kind of useless in
trying to figure out slope and

00:02:32.280 --> 00:02:33.130
y-intercept.

00:02:33.130 --> 00:02:35.200
I don't know what the slope and
y-intercept is when I look

00:02:35.200 --> 00:02:37.100
at it in standard form.

00:02:37.100 --> 00:02:39.490
My favorite is slope-intercept
form, because it tells you

00:02:39.490 --> 00:02:41.790
exactly the slope and
an intercept.

00:02:41.790 --> 00:02:45.140
Point-slope, easy to get to,
and you can look at it and

00:02:45.140 --> 00:02:46.560
figure out the slope.

00:02:46.560 --> 00:02:48.870
But y-intercept, you have to
do a little bit of work to

00:02:48.870 --> 00:02:49.630
figure it out.

00:02:49.630 --> 00:02:52.315
But at least you can just go
immediately from the slope and

00:02:52.315 --> 00:02:53.570
a point to it.

00:02:53.570 --> 00:02:56.240
But anyway, let's go from this
equation, which is written in

00:02:56.240 --> 00:03:00.360
point-slope form, and get
it to the standard form.

00:03:00.360 --> 00:03:02.640
So we want to get it to the
standard form, to the same

00:03:02.640 --> 00:03:04.250
type of standard form.

00:03:04.250 --> 00:03:07.150
So a good thing to do, let's
just distribute things out. y

00:03:07.150 --> 00:03:12.700
minus 7 is equal to negative 5
times x, negative 5x, plus

00:03:12.700 --> 00:03:18.370
negative 5, times negative
12, which is positive 60.

00:03:18.370 --> 00:03:20.760
Now, we want all of the variable
terms on the left,

00:03:20.760 --> 00:03:24.190
all of the constant terms
on the right.

00:03:24.190 --> 00:03:27.440
So let's add 7 to both sides
of this equation.

00:03:27.440 --> 00:03:31.440
So plus 7 to both sides
of this equation.

00:03:31.440 --> 00:03:32.800
What does it become?

00:03:32.800 --> 00:03:37.100
Well, the minus 7 disappears,
because negative 7 plus 7.

00:03:37.100 --> 00:03:41.920
So you're just left with
a y being equal to

00:03:41.920 --> 00:03:49.000
negative 5x plus 67.

00:03:49.000 --> 00:03:52.370
Now, if we want this x term on
the left-hand side, we could

00:03:52.370 --> 00:03:55.010
add 5x to both sides.

00:03:55.010 --> 00:04:02.660
So let's add 5x to both sides
of this equation.

00:04:02.660 --> 00:04:08.250
And we will get y plus 5x
is equal to-- these

00:04:08.250 --> 00:04:10.880
cancel out-- 67.

00:04:10.880 --> 00:04:12.620
Now, this is pretty much
standard form.

00:04:12.620 --> 00:04:15.230
If you really want to be a
stickler for it, you can

00:04:15.230 --> 00:04:16.579
rearrange these two.

00:04:16.579 --> 00:04:21.685
So it'd be 5x plus
y is equal to 67.

00:04:21.685 --> 00:04:23.390
And you are done.

00:04:23.390 --> 00:04:25.630
Let's do one more of these.

00:04:25.630 --> 00:04:28.950
So this is in neither
point-slope nor in

00:04:28.950 --> 00:04:30.070
slope-intercept form.

00:04:30.070 --> 00:04:31.920
It's just in some type
of intermediary

00:04:31.920 --> 00:04:33.470
mixed form right there.

00:04:33.470 --> 00:04:36.080
This looks like some type of
point-slope, but this looks

00:04:36.080 --> 00:04:37.060
like something different.

00:04:37.060 --> 00:04:38.710
So it's really not
point-slope.

00:04:38.710 --> 00:04:40.490
Let's see if we can
algebraically manipulate it to

00:04:40.490 --> 00:04:42.050
the standard form.

00:04:42.050 --> 00:04:45.570
So we get 3y plus 5.

00:04:45.570 --> 00:04:46.780
Let's distribute out this 4.

00:04:46.780 --> 00:04:50.950
So it's equal to 4x minus 36.

00:04:50.950 --> 00:04:52.920
Let's do exactly what
we did in the last.

00:04:52.920 --> 00:04:55.370
I'm using different notation on
purpose, to expose you to

00:04:55.370 --> 00:04:56.110
different things.

00:04:56.110 --> 00:04:59.320
So instead of doing it this way,
I'm going to subtract 5

00:04:59.320 --> 00:05:01.290
from both sides, but I'm going
to do it on the same line.

00:05:01.290 --> 00:05:05.440
So I'm going to subtract
5 from both sides.

00:05:05.440 --> 00:05:09.330
And so the left-hand side of
this equation becomes 3y,

00:05:09.330 --> 00:05:14.220
because these two guys cancel
out, and that is equal to 4x.

00:05:14.220 --> 00:05:18.290
And then what is minus
36 minus 5?

00:05:18.290 --> 00:05:21.470
That's minus 41.

00:05:21.470 --> 00:05:24.000
And now we want the x terms
of the left-hand side.

00:05:24.000 --> 00:05:29.330
So let's subtract 4x from both
sides of this equation.

00:05:29.330 --> 00:05:33.920
So negative 4x plus,
and then minus 4x.

00:05:33.920 --> 00:05:35.980
What does our equation become?

00:05:35.980 --> 00:05:41.330
Well, the left-hand side just
stays negative 4x plus 3y.

00:05:41.330 --> 00:05:44.400
And the right-hand, the reason
why we subtracted 4x is so it

00:05:44.400 --> 00:05:45.780
cancels out with that.

00:05:45.780 --> 00:05:48.800
You just have a negative 41.

00:05:48.800 --> 00:05:49.990
And we're done.

00:05:49.990 --> 00:05:52.970
We are in standard form.

00:05:52.970 --> 00:05:54.460
Now, let's go the other way.

00:05:54.460 --> 00:05:57.220
Let's start with some equations
in standard form and

00:05:57.220 --> 00:06:00.090
figure out their slope
and y-intercept.

00:06:00.090 --> 00:06:02.080
And the best way I know to
figure out the slope and

00:06:02.080 --> 00:06:05.460
y-intercept is to put it into
slope-intercept form.

00:06:05.460 --> 00:06:09.310
So we want to put these
equations right here into the

00:06:09.310 --> 00:06:12.820
form, y is equal to mx plus b.

00:06:12.820 --> 00:06:15.740
So we're essentially
solving for y.

00:06:15.740 --> 00:06:16.510
Let's do that.

00:06:16.510 --> 00:06:19.420
So the best thing to do here--
so let me rewrite it.

00:06:19.420 --> 00:06:26.860
5x minus 2y is equal to 15.

00:06:26.860 --> 00:06:29.680
Let's subtract 5x
from both sides.

00:06:29.680 --> 00:06:36.530
So minus 5x plus, you
have a minus 5x.

00:06:36.530 --> 00:06:37.690
These cancel out.

00:06:37.690 --> 00:06:40.310
And so you're left with
negative 2y is

00:06:40.310 --> 00:06:44.580
equal to 15 minus 5x.

00:06:44.580 --> 00:06:48.750
And now, let's divide everything
by negative 2.

00:06:48.750 --> 00:06:54.960
If you divide everything by
negative 2, what do we get?

00:06:54.960 --> 00:06:57.620
The left-hand side
just becomes a y.

00:06:57.620 --> 00:07:04.530
y is equal to-- 15 divided by
negative 2 is negative 7.5.

00:07:04.530 --> 00:07:07.310
And then negative 5 divided by
negative 2-- you can imagine

00:07:07.310 --> 00:07:10.540
I'm distributing the negative
1/2 if you will.

00:07:10.540 --> 00:07:12.410
I'm dividing both of these
by negative 2.

00:07:12.410 --> 00:07:16.955
So negative 5 divided by
negative 2 is positive 2.5x.

00:07:20.210 --> 00:07:21.820
And if you really wanted to put
it in the slope-intercept

00:07:21.820 --> 00:07:23.760
form, you could say that y is
equal to-- you could just

00:07:23.760 --> 00:07:29.030
rearrange these--
2.5x minus 7.5.

00:07:29.030 --> 00:07:30.430
You want the slope.

00:07:30.430 --> 00:07:32.710
It's right here.

00:07:32.710 --> 00:07:35.490
That is our slope.

00:07:35.490 --> 00:07:38.070
You want the y-intercept.

00:07:38.070 --> 00:07:39.690
Actually, let me be careful.

00:07:39.690 --> 00:07:42.110
It is right there.

00:07:42.110 --> 00:07:44.230
It is negative 7.5.

00:07:44.230 --> 00:07:45.520
That is the y-intercept.

00:07:45.520 --> 00:07:47.090
And now this would be a form
that's actually pretty

00:07:47.090 --> 00:07:49.650
straightforward to
graph it in.

00:07:49.650 --> 00:07:52.120
Let's do this one.

00:07:52.120 --> 00:07:54.320
So once again, we just
need to solve for y.

00:07:54.320 --> 00:07:56.990
So let's subtract 3x
from both sides.

00:07:56.990 --> 00:08:02.100
So you get 6y is equal
to 25 minus 3x.

00:08:02.100 --> 00:08:05.560
And then you can divide
both sides by 6.

00:08:05.560 --> 00:08:11.870
So you're left with y is equal
to 25 over 6 minus 3 over 6,

00:08:11.870 --> 00:08:14.060
or minus 1/2x.

00:08:14.060 --> 00:08:16.550
If you really want it in this
from, you just rearrange this.

00:08:16.550 --> 00:08:22.980
y is equal to negative
1/2x plus 25 over 6.

00:08:22.980 --> 00:08:24.610
Where is the slope?

00:08:24.610 --> 00:08:25.890
Here is the slope.

00:08:25.890 --> 00:08:28.670
Negative 1/2, that
is the slope.

00:08:28.670 --> 00:08:30.220
Where is the y-intercept?

00:08:30.220 --> 00:08:31.790
That's the y-intercept.

00:08:31.790 --> 00:08:33.030
That is our b.

00:08:33.030 --> 00:08:36.830
The point 0, 25 over
6 is on the line.

00:08:36.830 --> 00:08:38.240
Let's do one more of these.

00:08:41.929 --> 00:08:47.080
So we get 9x minus
9y is equal to 4.

00:08:47.080 --> 00:08:51.670
Just for fun, let's just start
off by dividing both sides of

00:08:51.670 --> 00:08:52.960
the equation by 9.

00:08:52.960 --> 00:08:54.080
You don't have to
do it that way.

00:08:54.080 --> 00:08:56.100
But this is kind of a fun way
to do it, because the

00:08:56.100 --> 00:08:58.460
coefficients here will
immediately become 1.

00:08:58.460 --> 00:09:03.250
So if you divide both sides of
the equation by 9, if you

00:09:03.250 --> 00:09:08.970
divide everything by 9, it
becomes-- actually, well,

00:09:08.970 --> 00:09:09.960
let's divide everything.

00:09:09.960 --> 00:09:14.530
Let's divide everything by
negative 9, even better.

00:09:14.530 --> 00:09:16.010
I'm just doing this for fun.

00:09:16.010 --> 00:09:19.140
So this first term will
become negative x.

00:09:19.140 --> 00:09:21.120
The second term, you have a
negative 9 divided by a

00:09:21.120 --> 00:09:23.620
negative 9, it will
be a plus y.

00:09:23.620 --> 00:09:25.760
And then this last term
will just become a

00:09:25.760 --> 00:09:28.800
negative 4 over 9.

00:09:28.800 --> 00:09:31.110
Actually, let me write
this out here.

00:09:31.110 --> 00:09:33.540
Negative 4 over 9.

00:09:33.540 --> 00:09:35.380
I'm giving some space there.

00:09:35.380 --> 00:09:38.270
Now, we want the x on the
right-hand side, so let's add

00:09:38.270 --> 00:09:39.836
x to both sides of
this equation.

00:09:44.150 --> 00:09:45.720
These cancel out.

00:09:45.720 --> 00:09:51.960
And then the equation becomes
y is equal to x minus 4/9.

00:09:51.960 --> 00:09:53.530
Where is the slope?

00:09:53.530 --> 00:09:55.460
The slope is the coefficient
on the x term.

00:09:55.460 --> 00:09:58.990
The slope is equal to 1.

00:09:58.990 --> 00:10:00.270
Where is the y-intercept?

00:10:00.270 --> 00:10:02.956
The y-intercept is
right there.

00:10:02.956 --> 00:10:06.040
It is negative 4/9.


Linear Equations in Point Slope Form

WEBVTT
Kind: captions
Language: en

00:00:00.380 --> 00:00:04.655
Let's do some equations of lines
in point-slope form.

00:00:07.540 --> 00:00:12.645
And this is different from y or
from slope-intercept form.

00:00:15.650 --> 00:00:18.200
But they really are just two
different ways of writing the

00:00:18.200 --> 00:00:19.330
same equation.

00:00:19.330 --> 00:00:21.220
We'll see that with a
couple of examples.

00:00:21.220 --> 00:00:24.450
And you might remember that
slope-intercept form were

00:00:24.450 --> 00:00:30.030
equations of the form, y is
equal to mx plus b-- we did

00:00:30.030 --> 00:00:33.280
this in the last video-- where
m is the slope, b is the

00:00:33.280 --> 00:00:34.850
intercept, the y-intercept.

00:00:34.850 --> 00:00:36.060
That's why it's called
slope-intercept.

00:00:36.060 --> 00:00:38.320
You have the slope and
the intercept.

00:00:38.320 --> 00:00:46.650
In point-slope form, it takes
the form y minus y1-- and I'll

00:00:46.650 --> 00:00:52.680
tell you what y1 is in a
second-- is equal to m times x

00:00:52.680 --> 00:01:00.590
minus x1, where the coordinate
x1, y1 is a point on the line.

00:01:00.590 --> 00:01:03.530
That's why this is called
point-slope form.

00:01:03.530 --> 00:01:06.440
Now, these are just two
different ways of writing the

00:01:06.440 --> 00:01:07.520
exact same thing.

00:01:07.520 --> 00:01:10.960
You can always algebraically
manipulate this to get that,

00:01:10.960 --> 00:01:13.870
or algebraically manipulate
that to get that.

00:01:13.870 --> 00:01:15.800
And I'll show you that with
a couple of examples.

00:01:15.800 --> 00:01:18.960
But let's just do a few in
point-slope form, just to make

00:01:18.960 --> 00:01:21.140
things concrete in your head.

00:01:21.140 --> 00:01:25.030
So here we have a line that has
a slope of negative 1 over

00:01:25.030 --> 00:01:29.160
10, so m is equal to
negative 1 over 10.

00:01:29.160 --> 00:01:32.460
And it goes through the
point 10 comma 2.

00:01:32.460 --> 00:01:36.320
So we can directly go
to point-slope form.

00:01:36.320 --> 00:01:37.430
So let's get a point.

00:01:37.430 --> 00:01:41.470
A point here is the
point 10 comma 2.

00:01:41.470 --> 00:01:45.470
So we can immediately go to
point-slope form. y minus,

00:01:45.470 --> 00:01:47.730
this is a y-value that's
on the line.

00:01:47.730 --> 00:01:53.730
So y minus 2 is going to be
equal to the slope, negative 1

00:01:53.730 --> 00:02:01.530
over 10 times x minus an
x-value, times x minus 10,

00:02:01.530 --> 00:02:02.020
just like that.

00:02:02.020 --> 00:02:02.780
And we're done.

00:02:02.780 --> 00:02:05.650
We just put it in point-slope
form.

00:02:05.650 --> 00:02:08.410
There's two things I want
to point out to you.

00:02:08.410 --> 00:02:09.610
One, why this makes sense.

00:02:09.610 --> 00:02:11.150
And I also want to show
you that this is

00:02:11.150 --> 00:02:12.300
equivalent to that.

00:02:12.300 --> 00:02:14.910
So the first thing is why
does this make sense?

00:02:14.910 --> 00:02:17.920
Well, all this is saying, if you
divide both sides of this

00:02:17.920 --> 00:02:24.710
equation by that right over
there, you get y2 over x minus

00:02:24.710 --> 00:02:27.880
10 is equal to negative
1 over 10.

00:02:27.880 --> 00:02:32.540
Or you get the change in y
between any point and 2, and

00:02:32.540 --> 00:02:36.380
the point 2, over
the change in x.

00:02:36.380 --> 00:02:40.400
So you get the change in y over
the change in x for any

00:02:40.400 --> 00:02:46.200
point x, y on that line,
relative to the point 10 comma

00:02:46.200 --> 00:02:48.690
2, is going to be negative
1 over 10.

00:02:48.690 --> 00:02:52.370
This is just the definition
of your slope.

00:02:52.370 --> 00:02:53.580
Hopefully I don't confuse
you there.

00:02:53.580 --> 00:02:55.720
I'm just showing you that this
is just using the definition

00:02:55.720 --> 00:02:59.190
of the slope to create the
equation of the line.

00:02:59.190 --> 00:03:01.140
Now, the other thing I want
to show is that this is

00:03:01.140 --> 00:03:03.100
completely equivalent to this.

00:03:03.100 --> 00:03:06.110
We can just algebraically
manipulate this to get that.

00:03:06.110 --> 00:03:07.060
And let's do that.

00:03:07.060 --> 00:03:10.280
So this right here is the
answer to the problem.

00:03:10.280 --> 00:03:12.340
But let's play around with
it algebraically to

00:03:12.340 --> 00:03:13.780
get it in that form.

00:03:13.780 --> 00:03:17.170
So if we get y minus 2 is equal
to-- let's distribute

00:03:17.170 --> 00:03:18.720
the negative 1 over 10.

00:03:18.720 --> 00:03:22.640
So it's negative 1 over 10 x,
and then negative 1 over 10

00:03:22.640 --> 00:03:26.020
times negative 10 is plus 1.

00:03:26.020 --> 00:03:28.990
Now we can add 2 to both
sides of this equation.

00:03:28.990 --> 00:03:34.850
And you get y is equal to
negative 1 over 10 x plus 3.

00:03:34.850 --> 00:03:37.370
So just algebraically
manipulating it, we were able

00:03:37.370 --> 00:03:39.830
to put it into slope-intercept
form.

00:03:39.830 --> 00:03:43.750
So these two things are
completely equivalent.

00:03:43.750 --> 00:03:47.680
Let's do a couple more problems.
The line contains

00:03:47.680 --> 00:03:50.630
the points 10 comma
12 and 5, 25.

00:03:50.630 --> 00:03:53.040
So let's figure out the slope.

00:03:53.040 --> 00:03:59.060
So the slope, which is equal to
change in y over change in

00:03:59.060 --> 00:04:03.000
x, is equal to-- well, let's
just use this point first. So

00:04:03.000 --> 00:04:19.870
let's say it's 12 minus
25 over 10 minus 5.

00:04:19.870 --> 00:04:25.760
This is going to be equal to--
12 minus 25 is negative 13,

00:04:25.760 --> 00:04:29.530
over 10 minus 5, which is 5.

00:04:29.530 --> 00:04:32.270
So the slope here is
negative 13 over 5.

00:04:32.270 --> 00:04:34.980
Let's put it in point-slope
form, the equation.

00:04:34.980 --> 00:04:39.040
So it's going to be y minus--
let's use this point right

00:04:39.040 --> 00:04:46.440
here-- y minus 25 is equal to
the slope negative 13 over 5

00:04:46.440 --> 00:04:51.690
times x minus this point, 5.

00:04:51.690 --> 00:04:56.350
We just knew that the point
5 comma 25 is on the line.

00:04:56.350 --> 00:05:00.890
So y minus 25 is equal to the
slope, which we figured out,

00:05:00.890 --> 00:05:02.850
times x minus 5.

00:05:02.850 --> 00:05:04.060
And we're done.

00:05:04.060 --> 00:05:04.650
That's all.

00:05:04.650 --> 00:05:07.580
If you want, out of interest,
you could do the algebra to

00:05:07.580 --> 00:05:10.470
put this into the
slope-intercept form, to the

00:05:10.470 --> 00:05:13.930
mx plus b form, and see that
they are completely

00:05:13.930 --> 00:05:16.460
equivalent.

00:05:16.460 --> 00:05:18.770
Let's do another one.

00:05:18.770 --> 00:05:20.050
So they gave us our slope.

00:05:20.050 --> 00:05:23.700
It's 3/5, and the y-intercept
is negative 3.

00:05:23.700 --> 00:05:25.945
So here, immediately, this would
be very easy to put it

00:05:25.945 --> 00:05:27.420
in the slope-intercept form.

00:05:35.020 --> 00:05:40.650
The equation of this line is y
is equal to the slope 3/5x,

00:05:40.650 --> 00:05:43.200
plus the y-intercept, minus 3.

00:05:43.200 --> 00:05:44.070
And we'd be done.

00:05:44.070 --> 00:05:47.590
But how do you put this
into point-slope form?

00:05:53.190 --> 00:05:54.830
Well, we know the slope.

00:05:54.830 --> 00:05:56.950
We know that m is
equal to 3/5.

00:05:59.670 --> 00:06:01.750
But do we know any points
on this line?

00:06:01.750 --> 00:06:04.150
You need a point and a slope
to immediately put it into

00:06:04.150 --> 00:06:05.660
point-slope form.

00:06:05.660 --> 00:06:07.180
Well, we know one point.

00:06:07.180 --> 00:06:08.300
We know the y-intercept.

00:06:08.300 --> 00:06:10.150
The y-intercept is negative 3.

00:06:10.150 --> 00:06:13.420
That means when x is equal to
0, y is equal to negative 3.

00:06:13.420 --> 00:06:16.520
So our point is, the point
0, negative 3.

00:06:16.520 --> 00:06:18.120
You could have tried to figure
out other points.

00:06:18.120 --> 00:06:20.450
You could have said, oh,
when x is equal to 5,

00:06:20.450 --> 00:06:21.500
y is equal to 0.

00:06:21.500 --> 00:06:24.090
There's all sorts of things
you could've tried out.

00:06:24.090 --> 00:06:26.500
But this one was just sitting
there for us.

00:06:26.500 --> 00:06:27.840
The y-intercept is negative 3.

00:06:27.840 --> 00:06:31.640
That means that the point 0,
minus 3 is on the line.

00:06:31.640 --> 00:06:34.180
So let's write it in
point-slope form.

00:06:34.180 --> 00:06:42.150
y minus this y-value, so y minus
negative 3, is equal to

00:06:42.150 --> 00:06:48.080
the slope 3/5 times x minus
this x-value, this

00:06:48.080 --> 00:06:52.000
x-coordinate, x minus 0.

00:06:52.000 --> 00:06:53.420
And this is point-slope form.

00:06:53.420 --> 00:06:58.810
This could be y plus 3 is equal
to 3/5 times-- we could

00:06:58.810 --> 00:07:00.200
write times x minus 0.

00:07:00.200 --> 00:07:03.610
If you really wanted to make it
look like point-slope form,

00:07:03.610 --> 00:07:05.510
this would be point-slope form,
but it's kind of silly

00:07:05.510 --> 00:07:07.080
to write x minus 0.

00:07:07.080 --> 00:07:09.936
So you could just write y
plus 3 is equal to 3/5x.

00:07:13.290 --> 00:07:15.380
It's not 100% clear that
you're in point-slope

00:07:15.380 --> 00:07:16.230
form yet right now.

00:07:16.230 --> 00:07:18.480
But I think you still need
to write x minus 0.

00:07:18.480 --> 00:07:21.870
And obviously, to go from here
to there, you just have to

00:07:21.870 --> 00:07:24.110
subtract 3 from both sides
and you'll get that.

00:07:24.110 --> 00:07:25.920
So these are almost
equivalent.

00:07:25.920 --> 00:07:28.720
I mean, they are equivalent in
terms of what they represent.

00:07:28.720 --> 00:07:30.750
They're even almost equivalent
in how you write them.

00:07:30.750 --> 00:07:33.300
You just have to subtract 3
from both sides of this

00:07:33.300 --> 00:07:36.350
equation to get to that one.

00:07:36.350 --> 00:07:37.770
Let's do another one.

00:07:37.770 --> 00:07:42.260
Let's put this in point-slope
form.

00:07:42.260 --> 00:07:44.880
They're giving us information.

00:07:44.880 --> 00:07:48.570
They're giving it in the
form of x, f of x.

00:07:48.570 --> 00:07:53.680
So in this situation, when
x is negative 7, f of

00:07:53.680 --> 00:07:55.860
x is equal to 5.

00:07:55.860 --> 00:07:59.580
And this coordinate, they're
telling us when x is equal to

00:07:59.580 --> 00:08:03.810
3, f of x is equal
to negative 4.

00:08:03.810 --> 00:08:08.220
So just like that, we can figure
out the slope first. So

00:08:08.220 --> 00:08:14.895
the slope, which is equal change
in y over change in x.

00:08:14.895 --> 00:08:15.950
So let's do this.

00:08:15.950 --> 00:08:29.130
It's negative 4 minus 5, over
3 minus negative 7.

00:08:29.130 --> 00:08:32.900
And this is going to be equal
to-- negative 4 minus 5 is

00:08:32.900 --> 00:08:34.460
negative 9.

00:08:34.460 --> 00:08:38.299
And then 3 minus negative 7,
that's the same thing as 3

00:08:38.299 --> 00:08:40.049
plus 7, that's 10.

00:08:40.049 --> 00:08:42.870
So slope is negative 9/10.

00:08:42.870 --> 00:08:43.830
And so to put this
in point-slope

00:08:43.830 --> 00:08:45.083
form is really easy.

00:08:52.000 --> 00:08:55.710
It's just going to be y--
let's do it this way-- y

00:08:55.710 --> 00:09:01.770
minus-- I'll color code it-- 3
is equal to-- I'll do this

00:09:01.770 --> 00:09:06.480
back to the green color-- is
equal to the slope, is equal

00:09:06.480 --> 00:09:16.415
to negative 9 over 10, times
this x minus this coordinate,

00:09:16.415 --> 00:09:20.180
x minus negative 4.

00:09:20.180 --> 00:09:22.090
Then we can close it.

00:09:22.090 --> 00:09:23.560
So this is in point-slope
form.

00:09:23.560 --> 00:09:26.050
We obviously can simplify this
negative a little bit.

00:09:26.050 --> 00:09:32.210
We can rewrite it as y minus
3 is equal to negative 9/10

00:09:32.210 --> 00:09:35.330
times x, plus 4.

00:09:35.330 --> 00:09:37.340
And we are done.


More examples of constructing linear equations in slope-intercept form

WEBVTT
Kind: captions
Language: en

00:00:00.450 --> 00:00:03.570
In this video I'm going to do a
bunch of examples of finding

00:00:03.570 --> 00:00:07.170
the equations of lines in
slope-intercept form.

00:00:07.170 --> 00:00:09.610
Just as a bit of a review, that
means equations of lines

00:00:09.610 --> 00:00:17.050
in the form of y is equal to mx
plus b where m is the slope

00:00:17.050 --> 00:00:21.200
and b is the y-intercept.

00:00:21.200 --> 00:00:24.870
So let's just do a bunch of
these problems. So here they

00:00:24.870 --> 00:00:28.900
tell us that a line has a slope
of negative 5, so m is

00:00:28.900 --> 00:00:30.740
equal to negative 5.

00:00:30.740 --> 00:00:34.290
And it has a y-intercept of 6.

00:00:34.290 --> 00:00:36.300
So b is equal to 6.

00:00:36.300 --> 00:00:37.985
So this is pretty
straightforward.

00:00:37.985 --> 00:00:41.530
The equation of this line
is y is equal to

00:00:41.530 --> 00:00:47.550
negative 5x plus 6.

00:00:47.550 --> 00:00:49.570
That wasn't too bad.

00:00:49.570 --> 00:00:51.570
Let's do this next
one over here.

00:00:51.570 --> 00:00:54.300
The line has a slope of negative
1 and contains the

00:00:54.300 --> 00:00:57.320
point 4/5 comma 0.

00:00:57.320 --> 00:01:00.600
So they're telling us the slope,
slope of negative 1.

00:01:00.600 --> 00:01:05.230
So we know that m is equal to
negative 1, but we're not 100%

00:01:05.230 --> 00:01:09.190
sure about where the y-intercept
is just yet.

00:01:09.190 --> 00:01:12.510
So we know that this equation
is going to be of the form y

00:01:12.510 --> 00:01:19.300
is equal to the slope negative
1x plus b, where b is the

00:01:19.300 --> 00:01:20.460
y-intercept.

00:01:20.460 --> 00:01:23.650
Now, we can use this coordinate
information, the

00:01:23.650 --> 00:01:25.870
fact that it contains this
point, we can use that

00:01:25.870 --> 00:01:28.590
information to solve for b.

00:01:28.590 --> 00:01:31.530
The fact that the line contains
this point means that

00:01:31.530 --> 00:01:37.690
the value x is equal to 4/5, y
is equal to 0 must satisfy

00:01:37.690 --> 00:01:38.265
this equation.

00:01:38.265 --> 00:01:43.120
So let's substitute those in.
y is equal to 0 when x is

00:01:43.120 --> 00:01:44.090
equal to 4/5.

00:01:44.090 --> 00:01:50.170
So 0 is equal to negative
1 times 4/5 plus b.

00:01:50.170 --> 00:01:52.810
I'll scroll down a little bit.

00:01:52.810 --> 00:01:58.110
So let's see, we get a 0 is
equal to negative 4/5 plus b.

00:01:58.110 --> 00:02:02.040
We can add 4/5 to both sides
of this equation.

00:02:02.040 --> 00:02:04.250
So we get add a 4/5 there.

00:02:04.250 --> 00:02:07.320
We could add a 4/5 to
that side as well.

00:02:07.320 --> 00:02:10.100
The whole reason I did that is
so that cancels out with that.

00:02:10.100 --> 00:02:12.130
You get b is equal to 4/5.

00:02:16.250 --> 00:02:19.180
So we now have the equation
of the line.

00:02:19.180 --> 00:02:23.040
y is equal to negative 1 times
x, which we write as negative

00:02:23.040 --> 00:02:32.500
x, plus b, which is 4/5,
just like that.

00:02:32.500 --> 00:02:34.480
Now we have this one.

00:02:34.480 --> 00:02:39.580
The line contains the point
2 comma 6 and 5 comma 0.

00:02:39.580 --> 00:02:42.540
So they haven't given us the
slope or the y-intercept

00:02:42.540 --> 00:02:43.030
explicitly.

00:02:43.030 --> 00:02:45.350
But we could figure out both
of them from these

00:02:45.350 --> 00:02:45.650
coordinates.

00:02:45.650 --> 00:02:48.270
So the first thing we can do
is figure out the slope.

00:02:48.270 --> 00:02:53.750
So we know that the slope m is
equal to change in y over

00:02:53.750 --> 00:02:58.100
change in x, which is equal to--
What is the change in y?

00:02:58.100 --> 00:02:59.490
Let's start with this
one right here.

00:02:59.490 --> 00:03:00.985
So we do 6 minus 0.

00:03:04.210 --> 00:03:05.070
Let me do it this way.

00:03:05.070 --> 00:03:10.410
So that's a 6-- I want to make
it color-coded-- minus 0.

00:03:10.410 --> 00:03:14.340
So 6 minus 0, that's
our change in y.

00:03:14.340 --> 00:03:24.190
Our change in x is 2
minus 2 minus 5.

00:03:24.190 --> 00:03:26.320
The reason why I color-coded
it is I wanted to show you

00:03:26.320 --> 00:03:30.890
when I used this y term first,
I used the 6 up here, that I

00:03:30.890 --> 00:03:33.380
have to use this x term
first as well.

00:03:33.380 --> 00:03:36.730
So I wanted to show you, this
is the coordinate 2 comma 6.

00:03:36.730 --> 00:03:38.590
This is the coordinate
5 comma 0.

00:03:38.590 --> 00:03:41.650
I couldn't have swapped
the 2 and the 5 then.

00:03:41.650 --> 00:03:45.030
Then I would have gotten the
negative of the answer.

00:03:45.030 --> 00:03:46.080
But what do we get here?

00:03:46.080 --> 00:03:51.210
This is equal to
6 minus 0 is 6.

00:03:51.210 --> 00:03:54.770
2 minus 5 is negative 3.

00:03:54.770 --> 00:03:58.910
So this becomes negative 6
over 3, which is the same

00:03:58.910 --> 00:04:01.310
thing as negative 2.

00:04:01.310 --> 00:04:02.250
So that's our slope.

00:04:02.250 --> 00:04:06.920
So, so far we know that the line
must be, y is equal to

00:04:06.920 --> 00:04:12.580
the slope-- I'll do that in
orange-- negative 2 times x

00:04:12.580 --> 00:04:15.160
plus our y-intercept.

00:04:15.160 --> 00:04:17.779
Now we can do exactly what we
did in the last problem.

00:04:17.779 --> 00:04:20.579
We can use one of these
points to solve for b.

00:04:20.579 --> 00:04:22.029
We can use either one.

00:04:22.029 --> 00:04:25.920
Both of these are on the line,
so both of these must satisfy

00:04:25.920 --> 00:04:26.900
this equation.

00:04:26.900 --> 00:04:29.800
I'll use the 5 comma 0 because
it's always nice when

00:04:29.800 --> 00:04:31.020
you have a 0 there.

00:04:31.020 --> 00:04:32.820
The math is a little
bit easier.

00:04:32.820 --> 00:04:34.510
So let's put the 5
comma 0 there.

00:04:34.510 --> 00:04:38.900
So y is equal to 0 when
x is equal to 5.

00:04:38.900 --> 00:04:43.820
So y is equal to 0 when you have
negative 2 times 5, when

00:04:43.820 --> 00:04:47.700
x is equal to 5 plus b.

00:04:47.700 --> 00:04:52.650
So you get 0 is equal
to -10 plus b.

00:04:52.650 --> 00:04:57.820
If you add 10 to both sides of
this equation, let's add 10 to

00:04:57.820 --> 00:05:00.680
both sides, these
two cancel out.

00:05:00.680 --> 00:05:03.970
You get b is equal to
10 plus 0 or 10.

00:05:03.970 --> 00:05:06.420
So you get b is equal to 10.

00:05:06.420 --> 00:05:07.935
Now we know the equation
for the line.

00:05:07.935 --> 00:05:14.110
The equation is y-- let me do it
in a new color-- y is equal

00:05:14.110 --> 00:05:22.280
to negative 2x plus b plus 10.

00:05:22.280 --> 00:05:23.470
We are done.

00:05:23.470 --> 00:05:24.720
Let's do another one of these.

00:05:28.180 --> 00:05:31.270
All right, the line contains
the points 3 comma 5 and

00:05:31.270 --> 00:05:32.890
negative 3 comma 0.

00:05:32.890 --> 00:05:36.380
Just like the last problem, we
start by figuring out the

00:05:36.380 --> 00:05:40.380
slope, which we will call m.

00:05:40.380 --> 00:05:44.830
It's the same thing as the rise
over the run, which is

00:05:44.830 --> 00:05:48.190
the same thing as the change
in y over the change in x.

00:05:48.190 --> 00:05:50.070
If you were doing this for your
homework, you wouldn't

00:05:50.070 --> 00:05:50.870
have to write all this.

00:05:50.870 --> 00:05:52.920
I just want to make sure that
you understand that these are

00:05:52.920 --> 00:05:55.150
all the same things.

00:05:55.150 --> 00:05:58.520
Then what is our change in
y over our change in x?

00:05:58.520 --> 00:06:02.280
This is equal to, let's start
with the side first. It's just

00:06:02.280 --> 00:06:03.980
to show you I could pick
either of these points.

00:06:03.980 --> 00:06:14.050
So let's say it's 0 minus
5 just like that.

00:06:14.050 --> 00:06:17.000
So I'm using this coordinate
first. I'm kind of viewing it

00:06:17.000 --> 00:06:19.770
as the endpoint.

00:06:19.770 --> 00:06:22.420
Remember when I first learned
this, I would always be

00:06:22.420 --> 00:06:24.160
tempted to do the x
in the numerator.

00:06:24.160 --> 00:06:25.990
No, you use the y's
in the numerator.

00:06:25.990 --> 00:06:28.470
So that's the second
of the coordinates.

00:06:28.470 --> 00:06:38.435
That is going to be over
negative 3 minus 3.

00:06:41.250 --> 00:06:44.370
This is the coordinate
negative 3, 0.

00:06:44.370 --> 00:06:46.420
This is the coordinate 3, 5.

00:06:46.420 --> 00:06:47.980
We're subtracting that.

00:06:47.980 --> 00:06:49.310
So what are we going to get?

00:06:49.310 --> 00:06:52.570
This is going to be equal to--
I'll do it in a neutral

00:06:52.570 --> 00:06:56.210
color-- this is going to be
equal to the numerator is

00:06:56.210 --> 00:07:02.010
negative 5 over negative 3
minus 3 is negative 6.

00:07:02.010 --> 00:07:03.650
So the negatives cancel out.

00:07:03.650 --> 00:07:05.930
You get 5/6.

00:07:05.930 --> 00:07:08.700
So we know that the equation is
going to be of the form y

00:07:08.700 --> 00:07:15.560
is equal to 5/6 x plus b.

00:07:15.560 --> 00:07:18.600
Now we can substitute one of
these coordinates in for b.

00:07:18.600 --> 00:07:19.440
So let's do.

00:07:19.440 --> 00:07:21.310
I always like to use the one
that has the 0 in it.

00:07:21.310 --> 00:07:33.270
So y is a zero when x is
negative 3 plus b.

00:07:33.270 --> 00:07:37.810
So all I did is I substituted
negative 3 for x, 0 for y.

00:07:37.810 --> 00:07:40.860
I know I can do that because
this is on the line.

00:07:40.860 --> 00:07:44.040
This must satisfy the equation
of the line.

00:07:44.040 --> 00:07:45.600
Let's solve for b.

00:07:45.600 --> 00:07:49.990
So we get zero is equal to, well
if we divide negative 3

00:07:49.990 --> 00:07:51.830
by 3, that becomes a 1.

00:07:51.830 --> 00:07:54.890
If you divide 6 by 3,
that becomes a 2.

00:07:54.890 --> 00:08:02.380
So it becomes negative
5/2 plus b.

00:08:02.380 --> 00:08:05.280
We could add 5/2 to both
sides of the equation,

00:08:05.280 --> 00:08:08.630
plus 5/2, plus 5/2.

00:08:08.630 --> 00:08:10.850
I like to change my notation
just so you get

00:08:10.850 --> 00:08:12.520
familiar with both.

00:08:12.520 --> 00:08:17.800
So the equation becomes 5/2 is
equal to-- that's a 0-- is

00:08:17.800 --> 00:08:19.600
equal to b.

00:08:19.600 --> 00:08:22.090
b is 5/2.

00:08:22.090 --> 00:08:31.940
So the equation of our line is
y is equal to 5/6 x plus b,

00:08:31.940 --> 00:08:37.820
which we just figured out
is 5/2, plus 5/2.

00:08:37.820 --> 00:08:38.710
We are done.

00:08:38.710 --> 00:08:41.280
Let's do another one.

00:08:41.280 --> 00:08:43.500
We have a graph here.

00:08:43.500 --> 00:08:45.300
Let's figure out the equation
of this graph.

00:08:45.300 --> 00:08:46.900
This is actually, on some level,
a little bit easier.

00:08:46.900 --> 00:08:47.740
What's the slope?

00:08:47.740 --> 00:08:52.250
Slope is change in y
over change it x.

00:08:52.250 --> 00:08:53.310
So let's see what happens.

00:08:53.310 --> 00:08:57.900
When we move in x, when our
change in x is 1, so that is

00:08:57.900 --> 00:08:58.940
our change in x.

00:08:58.940 --> 00:09:00.850
So change in x is 1.

00:09:00.850 --> 00:09:04.130
I'm just deciding to change
my x by 1, increment by 1.

00:09:04.130 --> 00:09:05.900
What is the change in y?

00:09:05.900 --> 00:09:10.390
It looks like y changes
exactly by 4.

00:09:10.390 --> 00:09:14.980
It looks like my delta y, my
change in y, is equal to 4

00:09:14.980 --> 00:09:20.690
when my delta x is equal to 1.

00:09:20.690 --> 00:09:24.340
So change in y over change in
x, change in y is 4 when

00:09:24.340 --> 00:09:26.220
change in x is 1.

00:09:26.220 --> 00:09:30.380
So the slope is equal to 4.

00:09:30.380 --> 00:09:32.190
Now what's its y-intercept?

00:09:32.190 --> 00:09:33.720
Well here we can just
look at the graph.

00:09:33.720 --> 00:09:38.260
It looks like it intersects
y-axis at y is equal to

00:09:38.260 --> 00:09:41.600
negative 6, or at the
point 0, negative 6.

00:09:41.600 --> 00:09:44.180
So we know that b is equal
to negative 6.

00:09:46.950 --> 00:09:48.875
So we know the equation
of the line.

00:09:48.875 --> 00:09:56.630
The equation of the line is y is
equal to the slope times x

00:09:56.630 --> 00:09:59.030
plus the y-intercept.

00:09:59.030 --> 00:10:01.850
I should write that.

00:10:01.850 --> 00:10:07.840
So minus 6, that is plus
negative 6 So that is the

00:10:07.840 --> 00:10:09.800
equation of our line.

00:10:09.800 --> 00:10:12.980
Let's do one more of these.

00:10:12.980 --> 00:10:17.170
So they tell us that f of
1.5 is negative 3, f of

00:10:17.170 --> 00:10:18.750
negative 1 is 2.

00:10:18.750 --> 00:10:19.970
What is that?

00:10:19.970 --> 00:10:23.830
Well, all this is just a fancy
way of telling you that the

00:10:23.830 --> 00:10:30.530
point when x is 1.5, when you
put 1.5 into the function, the

00:10:30.530 --> 00:10:33.490
function evaluates
as negative 3.

00:10:33.490 --> 00:10:36.750
So this tells us that the
coordinate 1.5, negative 3 is

00:10:36.750 --> 00:10:38.270
on the line.

00:10:38.270 --> 00:10:41.960
Then this tells us that the
point when x is negative 1, f

00:10:41.960 --> 00:10:44.420
of x is equal to 2.

00:10:44.420 --> 00:10:47.540
This is just a fancy way of
saying that both of these two

00:10:47.540 --> 00:10:51.400
points are on the line,
nothing unusual.

00:10:51.400 --> 00:10:54.380
I think the point of this
problem is to get you familiar

00:10:54.380 --> 00:10:56.870
with function notation, for you
to not get intimidated if

00:10:56.870 --> 00:10:57.970
you see something like this.

00:10:57.970 --> 00:11:01.540
If you evaluate the function
at 1.5, you get negative 3.

00:11:01.540 --> 00:11:04.440
So that's the coordinate if
you imagine that y is

00:11:04.440 --> 00:11:06.020
equal to f of x.

00:11:06.020 --> 00:11:06.950
So this would be the
y-coordinate.

00:11:06.950 --> 00:11:09.250
It would be equal to negative
3 when x is 1.5.

00:11:09.250 --> 00:11:10.840
Anyway, I've said it
multiple times.

00:11:10.840 --> 00:11:13.280
Let's figure out the
slope of this line.

00:11:13.280 --> 00:11:20.020
The slope which is change in y
over change in x is equal to,

00:11:20.020 --> 00:11:27.460
let's start with 2 minus this
guy, negative 3-- these are

00:11:27.460 --> 00:11:32.880
the y-values-- over, all
of that over, negative

00:11:32.880 --> 00:11:40.140
1 minus this guy.

00:11:40.140 --> 00:11:43.330
Let me write it this way,
negative 1 minus

00:11:43.330 --> 00:11:48.440
that guy, minus 1.5.

00:11:48.440 --> 00:11:50.340
I do the colors because I want
to show you that the negative

00:11:50.340 --> 00:11:54.060
1 and the 2 are both coming from
this, that's why I use

00:11:54.060 --> 00:11:57.500
both of them first. If I used
these guys first, I would have

00:11:57.500 --> 00:12:00.495
to use both the x and the y
first. If I use the 2 first, I

00:12:00.495 --> 00:12:02.080
have to use the negative
1 first. That's why I'm

00:12:02.080 --> 00:12:03.390
color-coding it.

00:12:03.390 --> 00:12:08.360
So this is going to be equal
to 2 minus negative 3.

00:12:08.360 --> 00:12:10.370
That's the same thing
as 2 plus 3.

00:12:10.370 --> 00:12:11.620
So that is 5.

00:12:16.480 --> 00:12:20.040
Negative 1 minus 1.5
is negative 2.5.

00:12:23.830 --> 00:12:27.770
5 divided by 2.5
is equal to 2.

00:12:27.770 --> 00:12:30.250
So the slope of this
line is negative 2.

00:12:30.250 --> 00:12:32.130
Actually I'll take a little
aside to show you it doesn't

00:12:32.130 --> 00:12:34.480
matter what order
I do this in.

00:12:34.480 --> 00:12:36.180
If I use this coordinate first,
then I have to use that

00:12:36.180 --> 00:12:38.140
coordinate first. Let's
do it the other way.

00:12:38.140 --> 00:12:54.180
If I did it as negative 3
minus 2 over 1.5 minus

00:12:54.180 --> 00:12:59.810
negative 1, this should be minus
the 2 over 1.5 minus the

00:12:59.810 --> 00:13:01.060
negative 1.

00:13:03.300 --> 00:13:04.780
This should give me
the same answer.

00:13:04.780 --> 00:13:06.130
This is equal to what?

00:13:06.130 --> 00:13:12.860
Negative 3 minus 2 is negative
5 over 1.5 minus negative 1.

00:13:12.860 --> 00:13:14.520
That's 1.5 plus 1.

00:13:14.520 --> 00:13:16.610
That's over 2.5.

00:13:16.610 --> 00:13:18.840
So once again, this is
equal the negative 2.

00:13:18.840 --> 00:13:20.340
So I just wanted to show you,
it doesn't matter which one

00:13:20.340 --> 00:13:23.090
you pick as the starting or
the endpoint, as long as

00:13:23.090 --> 00:13:23.980
you're consistent.

00:13:23.980 --> 00:13:26.650
If this is the starting y,
this is the starting x.

00:13:26.650 --> 00:13:28.370
If this is the finishing
y, this has to be

00:13:28.370 --> 00:13:29.500
the finishing x.

00:13:29.500 --> 00:13:33.100
But anyway, we know that the
slope is negative 2.

00:13:33.100 --> 00:13:36.540
So we know the equation is y is
equal to negative 2x plus

00:13:36.540 --> 00:13:39.170
some y-intercept.

00:13:39.170 --> 00:13:40.720
Let's use one of these
coordinates.

00:13:40.720 --> 00:13:43.430
I'll use this one since it
doesn't have a decimal in it.

00:13:43.430 --> 00:13:47.450
So we know that y
is equal to 2.

00:13:47.450 --> 00:13:52.630
So y is equal to 2 when x
is equal to negative 1.

00:13:55.140 --> 00:13:57.290
Of course you have
your plus b.

00:13:57.290 --> 00:14:02.710
So 2 is equal to negative 2
times negative 1 is 2 plus b.

00:14:02.710 --> 00:14:06.390
If you subtract 2 from both
sides of this equation, minus

00:14:06.390 --> 00:14:10.370
2, minus 2, you're subtracting
it from both sides of this

00:14:10.370 --> 00:14:12.480
equation, you're going to get
0 on the left-hand side is

00:14:12.480 --> 00:14:14.520
equal to b.

00:14:14.520 --> 00:14:15.670
So b is 0.

00:14:15.670 --> 00:14:18.430
So the equation of our
line is just y is

00:14:18.430 --> 00:14:19.680
equal to negative 2x.

00:14:22.040 --> 00:14:23.870
Actually if you wanted to write
it in function notation,

00:14:23.870 --> 00:14:28.190
it would be that f of x is
equal to negative 2x.

00:14:28.190 --> 00:14:30.810
I kind of just assumed that
y is equal to f of x.

00:14:30.810 --> 00:14:32.420
But this is really
the equation.

00:14:32.420 --> 00:14:33.990
They never mentioned y's here.

00:14:33.990 --> 00:14:37.890
So you could just write f of x
is equal to 2x right here.

00:14:37.890 --> 00:14:40.190
Each of these coordinates
are the coordinates

00:14:40.190 --> 00:14:42.610
of x and f of x.

00:14:46.960 --> 00:14:49.960
So you could even view the
definition of slope as change

00:14:49.960 --> 00:14:53.320
in f of x over change in x.

00:14:53.320 --> 00:14:57.090
These are all equivalent ways
of viewing the same thing.


Word Problem Solving 3

WEBVTT
Kind: captions
Language: en

00:00:00.450 --> 00:00:03.420
Let's do some more practice
word problems.

00:00:03.420 --> 00:00:06.600
Patricia is building a sand
box for her daughter.

00:00:06.600 --> 00:00:11.470
It is to be 5 feet wide
and 8 feet long.

00:00:11.470 --> 00:00:13.140
So let me draw that.

00:00:13.140 --> 00:00:17.740
So the 5 feet wide-- it
is to be 5 feet wide.

00:00:17.740 --> 00:00:18.330
So there you go.

00:00:18.330 --> 00:00:20.690
That's 5 feet right there.

00:00:20.690 --> 00:00:22.640
And 8 feet long.

00:00:22.640 --> 00:00:23.500
I'll do that in orange.

00:00:23.500 --> 00:00:25.490
8 feet long.

00:00:25.490 --> 00:00:26.850
So this is like if we're
looking down

00:00:26.850 --> 00:00:27.770
into the sand box.

00:00:27.770 --> 00:00:30.860
8 feet long.

00:00:30.860 --> 00:00:33.100
So the sand box, if we're
looking down on it, would look

00:00:33.100 --> 00:00:33.810
something like that.

00:00:33.810 --> 00:00:35.120
That would be 5 feet.

00:00:35.120 --> 00:00:38.210
And this side over here
would also be 8 feet.

00:00:38.210 --> 00:00:40.980
All right, she wants the height
of the sand box to be 4

00:00:40.980 --> 00:00:44.130
inches above the height
of the sand.

00:00:44.130 --> 00:00:44.500
OK.

00:00:44.500 --> 00:00:47.870
So if we were to draw the sand
box like this-- so if you were

00:00:47.870 --> 00:00:52.200
to draw it-- so that's 8.

00:00:52.200 --> 00:00:55.310
That distance right
there is 5 feet.

00:00:55.310 --> 00:01:00.400
And then there is some height,
which I think we're going to

00:01:00.400 --> 00:01:02.320
have to figure out.

00:01:02.320 --> 00:01:04.750
Let me draw the whole
box just like that.

00:01:09.160 --> 00:01:10.820
The sand is going to
fill a certain

00:01:10.820 --> 00:01:13.380
amount of the sand box.

00:01:13.380 --> 00:01:18.250
So the sand is going to fill
maybe until right over there.

00:01:18.250 --> 00:01:22.180
So that's how much the sand is
going to fill of the sand box.

00:01:22.180 --> 00:01:24.040
All of that is sand.

00:01:24.040 --> 00:01:27.340
And she wants the height of the
sand box to be 4 inches

00:01:27.340 --> 00:01:29.000
above the height of the sand.

00:01:29.000 --> 00:01:32.010
So she wants this distance right
here, the amount that

00:01:32.010 --> 00:01:37.350
the height goes above the sand,
itself, to be 4 inches--

00:01:37.350 --> 00:01:39.440
or a third of a foot, depending
how you want to

00:01:39.440 --> 00:01:40.760
think about it.

00:01:40.760 --> 00:01:44.460
Now she has 30 cubic
feet of sand.

00:01:44.460 --> 00:01:47.220
She has 30 cubic feet of sand.

00:01:47.220 --> 00:01:51.810
So this sand right here
is 30 feet cubed.

00:01:51.810 --> 00:01:54.890
How high should the
sand box be?

00:01:54.890 --> 00:01:57.370
So this information right here,
the volume of the sand,

00:01:57.370 --> 00:02:01.100
tells us, using it, we can
figure out how high the level

00:02:01.100 --> 00:02:03.520
of the sand will be.

00:02:03.520 --> 00:02:05.130
Let's do it this way.

00:02:05.130 --> 00:02:10.600
Let's call l-- let
me just use x.

00:02:10.600 --> 00:02:13.200
Let's say x is the height
of the sand.

00:02:13.200 --> 00:02:19.200
x is equal to height of sand.

00:02:19.200 --> 00:02:22.990
So the total volume of our sand
is going to be the height

00:02:22.990 --> 00:02:26.600
of the sand times the
length of the box.

00:02:26.600 --> 00:02:29.980
Times 8 times the width
of the box.

00:02:29.980 --> 00:02:31.440
Times 5.

00:02:31.440 --> 00:02:33.590
And this is going to be the
total volume which they told

00:02:33.590 --> 00:02:36.430
us is 30 cubic feet.

00:02:36.430 --> 00:02:39.560
It is 30 cubic feet.

00:02:39.560 --> 00:02:43.520
We're taking x feet times
8 feet times 5 feet.

00:02:43.520 --> 00:02:45.900
So we'll get 30 feet cubed
if we were to keep

00:02:45.900 --> 00:02:47.380
track of the units.

00:02:47.380 --> 00:02:48.050
So what is this?

00:02:48.050 --> 00:02:52.580
8 times 5 is 40.

00:02:52.580 --> 00:02:55.250
So we get 40x is equal to 30.

00:02:55.250 --> 00:02:59.440
If we divide both sides by
40-- both sides of the

00:02:59.440 --> 00:03:01.470
equation by 40--
let me do that.

00:03:01.470 --> 00:03:05.730
You divide both sides by 40, you
get x is equal to-- that

00:03:05.730 --> 00:03:06.565
just becomes an x.

00:03:06.565 --> 00:03:13.660
X is equal to 30/40, where
it equals 3/4 of a foot.

00:03:13.660 --> 00:03:18.730
So this right here is going
to be 3/4 of a foot.

00:03:18.730 --> 00:03:21.440
Now what is 3/4 of a foot?

00:03:21.440 --> 00:03:23.650
Let's see, 3/4 of a foot.

00:03:23.650 --> 00:03:31.710
We know that there are
12 inches per foot.

00:03:31.710 --> 00:03:33.280
The feet cancel out.

00:03:33.280 --> 00:03:36.120
12 times 3/4 is what?

00:03:36.120 --> 00:03:39.050
You can divide the 12
by 4 and get a 3.

00:03:39.050 --> 00:03:40.770
Divide the 4 by 4
and you get a 1.

00:03:40.770 --> 00:03:42.040
You get 3 times 3.

00:03:42.040 --> 00:03:45.040
This is 9 inches.

00:03:45.040 --> 00:03:45.790
9 inches.

00:03:45.790 --> 00:03:50.800
So the height of the sand
right there is 9 inches.

00:03:50.800 --> 00:03:53.430
Now she wants the height of the
sand box to be 4 inches

00:03:53.430 --> 00:03:55.570
above the height of the sand.

00:03:55.570 --> 00:03:56.500
So this is 9 inches.

00:03:56.500 --> 00:03:58.320
She wants 4 more inches.

00:03:58.320 --> 00:04:04.220
So what she wants the height of
the sand box-- it should be

00:04:04.220 --> 00:04:07.250
the 9 inches of the height of
the sand plus 4, which is

00:04:07.250 --> 00:04:10.740
equal to 13 inches.

00:04:10.740 --> 00:04:14.370
And we are done.

00:04:14.370 --> 00:04:17.490
Let's do the next problem.

00:04:17.490 --> 00:04:22.130
It was a sale day in Macy's,
and everything was 20% less

00:04:22.130 --> 00:04:23.760
than the regular price.

00:04:23.760 --> 00:04:25.550
Peter bought a pair of shoes.

00:04:25.550 --> 00:04:29.145
And using a coupon, he got an
additional 10%-- let me do

00:04:29.145 --> 00:04:29.960
that in a different color.

00:04:29.960 --> 00:04:33.960
He got an additional 10% off
the discounted price.

00:04:33.960 --> 00:04:36.700
The price he paid was $36.

00:04:36.700 --> 00:04:39.170
The price he paid for the
shoes-- let me do that in a

00:04:39.170 --> 00:04:45.510
different color-- the price he
paid for the shoes was $36.

00:04:45.510 --> 00:04:47.920
How much did it originally
cost?

00:04:47.920 --> 00:04:56.300
So let's just use-- so x is
equal to original cost. There

00:04:56.300 --> 00:04:58.200
was just a regular
sale on that day.

00:04:58.200 --> 00:05:02.280
Everything was 20% less than
the regular price.

00:05:02.280 --> 00:05:05.190
So using that information, on
that day, if you wanted to buy

00:05:05.190 --> 00:05:12.280
the shoes, the cost of the shoes
would be x minus 0.20x,

00:05:12.280 --> 00:05:16.730
which is equal to 0.8x, right?

00:05:16.730 --> 00:05:19.490
If you have 20% off of a price,
you're going to be

00:05:19.490 --> 00:05:22.120
paying 80% of the
original price.

00:05:22.120 --> 00:05:24.410
So this is the price that
everyone would have paid

00:05:24.410 --> 00:05:25.640
without a coupon.

00:05:25.640 --> 00:05:28.270
And that's what that line is
telling us right there.

00:05:28.270 --> 00:05:31.180
Now Peter got a pair of shoes,
and using a coupon, got an

00:05:31.180 --> 00:05:34.090
additional 10% off of this.

00:05:34.090 --> 00:05:37.820
An additional 10% off of that.

00:05:37.820 --> 00:05:40.670
So he's paying 10%
less than this.

00:05:40.670 --> 00:05:44.650
Or another way to think of it
is, he only had to pay 90% of

00:05:44.650 --> 00:05:46.640
this price.

00:05:46.640 --> 00:05:54.510
If you take 0.8x and from that
you subtract 10% of this-- so

00:05:54.510 --> 00:06:03.760
from that, you subtract 10% of
that-- of 0.8x, this is the

00:06:03.760 --> 00:06:04.740
same thing.

00:06:04.740 --> 00:06:04.870
All right.

00:06:04.870 --> 00:06:06.290
This is a 1 time 0.8x.

00:06:09.180 --> 00:06:11.960
This is the same thing
as 1 minus 0.1.

00:06:11.960 --> 00:06:13.000
So this is the same thing.

00:06:13.000 --> 00:06:22.230
This is equal to 90% of 0.8x.

00:06:22.230 --> 00:06:23.830
And it might actually be
very natural for you to

00:06:23.830 --> 00:06:25.290
go straight to this.

00:06:25.290 --> 00:06:26.790
It's natural for me.

00:06:26.790 --> 00:06:29.930
If I'm paying 20% less than the
regular price, that means

00:06:29.930 --> 00:06:32.660
I'm going to pay 80% of
the regular price.

00:06:32.660 --> 00:06:35.740
And if I get an additional 10%
off of that, that means I'm

00:06:35.740 --> 00:06:38.720
only going to pay 90% of that.

00:06:38.720 --> 00:06:40.620
So this is what he's
going to pay.

00:06:40.620 --> 00:06:44.400
And they tell us that
price is $36.

00:06:44.400 --> 00:06:50.650
That price is equal to $36.

00:06:50.650 --> 00:06:53.170
So if we want to figure
out x, well, we could

00:06:53.170 --> 00:06:55.600
multiply 0.9 times 0.8.

00:06:55.600 --> 00:06:56.820
Let's do that.

00:06:56.820 --> 00:07:01.240
So 0.9 times 0.8.

00:07:01.240 --> 00:07:03.740
9 times 8 is 72.

00:07:03.740 --> 00:07:05.470
We have two numbers behind
the decimal.

00:07:05.470 --> 00:07:06.550
One, two.

00:07:06.550 --> 00:07:08.350
So it's 0.72.

00:07:08.350 --> 00:07:15.490
So you get 0.72x
is equal to 36.

00:07:15.490 --> 00:07:24.040
Or maybe we could say 72/100
x is equal to 36.

00:07:24.040 --> 00:07:26.620
Or we could even say this is
same thing as-- what is it?

00:07:26.620 --> 00:07:29.290
That's the same thing
as 36/50, right?

00:07:29.290 --> 00:07:34.330
36/50 x is equal to 36.

00:07:34.330 --> 00:07:37.410
And this isn't even in lowest
common forms. This is the same

00:07:37.410 --> 00:07:39.900
thing as 18/25.

00:07:39.900 --> 00:07:42.940
But I like this number because
I have a 36 here.

00:07:42.940 --> 00:07:45.800
And so I can multiply both sides
by the inverse of this.

00:07:45.800 --> 00:07:48.955
50/36 times this.

00:07:48.955 --> 00:07:52.830
I have to do to both sides
of the equation.

00:07:52.830 --> 00:07:53.900
36's cancel out.

00:07:53.900 --> 00:07:55.480
50's cancel out.

00:07:55.480 --> 00:07:57.010
36's cancel out.

00:07:57.010 --> 00:08:00.710
And you're left with
x is equal to $50.

00:08:00.710 --> 00:08:04.960
The shoes originally cost $50.

00:08:04.960 --> 00:08:07.000
Next question.

00:08:07.000 --> 00:08:11.160
Rosa has saved all year, and
wishes to spend the money she

00:08:11.160 --> 00:08:14.050
has on new clothes
and a vacation.

00:08:14.050 --> 00:08:18.360
She will spend 30% more on the
vacation then on the clothes.

00:08:18.360 --> 00:08:28.770
So let's say, let's say that C
is amount spent on clothes.

00:08:28.770 --> 00:08:32.070
She's going to spend 30%
more on the vacation.

00:08:32.070 --> 00:08:40.830
So let's say V is equal to the
amount spent on vacation.

00:08:40.830 --> 00:08:43.010
Well, they just told us that the
vacation-- she's going to

00:08:43.010 --> 00:08:45.030
spend 30% more than
on the clothes.

00:08:45.030 --> 00:08:48.660
So the vacation is going to be
equal to the amount she spends

00:08:48.660 --> 00:08:53.090
on the clothes plus 30%
of the amount she

00:08:53.090 --> 00:08:54.390
spends on the clothes.

00:08:54.390 --> 00:08:57.790
And this is the same
thing as 1.3.

00:08:57.790 --> 00:09:00.900
1.3 times the amount she
spends on the clothes.

00:09:00.900 --> 00:09:04.000
That's how much she's going
to spend on the vacation.

00:09:04.000 --> 00:09:07.930
Now, if she saved $1,000 in
total, how much money, to the

00:09:07.930 --> 00:09:10.230
nearest whole dollar, can she
spent on the vacation?

00:09:10.230 --> 00:09:14.080
So between V and C, she
can only spend $1,000.

00:09:14.080 --> 00:09:22.640
So V, vacation, plus her clothes
can only be $1,000.

00:09:22.640 --> 00:09:24.900
So we know what the vacation
is in terms of the clothes.

00:09:24.900 --> 00:09:25.910
We can do a little
substitution.

00:09:25.910 --> 00:09:29.240
The vacation is going to be
1.3 times the clothes.

00:09:29.240 --> 00:09:30.420
So we can substitute there.

00:09:30.420 --> 00:09:33.020
So you have 1.3 times
the clothes.

00:09:33.020 --> 00:09:35.660
That's the vacation expenditure,
plus the amount

00:09:35.660 --> 00:09:39.570
she spends on the clothes
is equal to 1,000.

00:09:39.570 --> 00:09:46.710
This is 1.3 plus 1 is 2.3, times
the amount she spends on

00:09:46.710 --> 00:09:49.500
clothes will be equal
to $1,000.

00:09:49.500 --> 00:09:53.330
Divide both sides by 2.3.

00:09:53.330 --> 00:09:55.890
I'll get the calculator
out for this one.

00:09:55.890 --> 00:10:07.770
So we have 1,000 divided by
2.3 is equal to $434.78.

00:10:07.770 --> 00:10:12.740
So C is equal to-- I already
forgot the number-- 434.78.

00:10:12.740 --> 00:10:17.080
$434.78.

00:10:17.080 --> 00:10:19.320
So if that's the amount she
spends on clothes, the rest of

00:10:19.320 --> 00:10:22.150
her money is going to be
spent on a vacation.

00:10:22.150 --> 00:10:24.770
So her vacation is going
to be 1,000 minus this.

00:10:24.770 --> 00:10:26.390
So her vacation expenditure
is going to

00:10:26.390 --> 00:10:31.710
be 1,000 minus $434.78.

00:10:31.710 --> 00:10:34.050
I'll do the calculator
for this one.

00:10:34.050 --> 00:10:34.410
Let's see.

00:10:34.410 --> 00:10:36.940
Let's make this into
a negative number.

00:10:36.940 --> 00:10:42.190
And then, let's add
1,000 to it.

00:10:42.190 --> 00:10:51.330
And we get $565.22.

00:10:51.330 --> 00:10:53.990
And I'll leave it up
to you to check it.

00:10:53.990 --> 00:10:56.090
This is how much she spends
on the vacation.

00:10:56.090 --> 00:10:58.060
These add up to $1,000.

00:10:58.060 --> 00:11:02.750
And this should be exactly 30%
more than this amount, the

00:11:02.750 --> 00:11:05.560
amount she spends
on her clothes.


Percent Problems

WEBVTT
Kind: captions
Language: en

00:00:00.560 --> 00:00:03.950
Let's do a couple of warm up
problems converting fractions

00:00:03.950 --> 00:00:05.540
to percentages and
then converting

00:00:05.540 --> 00:00:07.240
percentages to fractions.

00:00:07.240 --> 00:00:09.960
And then we can do some
actual word problems.

00:00:09.960 --> 00:00:17.860
So on our first warm up, let's
convert 5/24 to a percentage.

00:00:17.860 --> 00:00:19.710
And the way I like to do it,
I like to convert it to a

00:00:19.710 --> 00:00:22.250
decimal first. And once you have
it in the decimal form,

00:00:22.250 --> 00:00:25.530
it's pretty straight forward to
convert it to a percentage.

00:00:25.530 --> 00:00:28.050
So the way you convert to a
decimal is you divide the

00:00:28.050 --> 00:00:29.710
denominator into
the numerator.

00:00:29.710 --> 00:00:33.840
This can, literally, be read
as 5 divided by 24.

00:00:33.840 --> 00:00:35.430
So let's do a little
bit of division.

00:00:35.430 --> 00:00:39.325
24 goes into 5.

00:00:39.325 --> 00:00:42.530
And we're going to have to add
some extra spaces to the 5

00:00:42.530 --> 00:00:44.700
because, obviously, this is
going to be less than 1.

00:00:44.700 --> 00:00:47.595
24 is a larger number then 5.

00:00:47.595 --> 00:00:50.260
So let's put our decimal
right there.

00:00:50.260 --> 00:00:53.290
And we can now do
some division.

00:00:53.290 --> 00:00:57.470
24 goes in to 5 zero times.

00:00:57.470 --> 00:01:00.960
0 times 24 is 0.

00:01:00.960 --> 00:01:03.690
5 minus 0 is 5.

00:01:03.690 --> 00:01:07.690
Bring down this 0 right here.

00:01:07.690 --> 00:01:09.430
24 goes into 50.

00:01:09.430 --> 00:01:12.400
Well it goes into
that two times.

00:01:12.400 --> 00:01:17.460
2 times 24 is-- 2
times 4 is 8.

00:01:17.460 --> 00:01:20.220
2 times 2 is 4.

00:01:20.220 --> 00:01:21.570
Now we subtract.

00:01:21.570 --> 00:01:24.200
50 minus 48 is 2.

00:01:24.200 --> 00:01:26.260
Bring down another 0.

00:01:26.260 --> 00:01:28.210
We're going to keep
bringing down 0's

00:01:28.210 --> 00:01:29.800
until we have no remainder.

00:01:29.800 --> 00:01:32.080
So bring down another 0.

00:01:32.080 --> 00:01:36.570
24 goes into 20 zero times.

00:01:36.570 --> 00:01:39.840
0 times 24 is 0.

00:01:39.840 --> 00:01:42.590
20 minus 0 is 20.

00:01:42.590 --> 00:01:45.470
Let's bring down another 0.

00:01:45.470 --> 00:01:49.200
Let's bring down another 0.

00:01:49.200 --> 00:01:58.050
24 goes into 200-- let's
see, eight times?

00:01:58.050 --> 00:01:59.170
It'd be nine times?

00:01:59.170 --> 00:02:00.200
It's 180.

00:02:00.200 --> 00:02:01.480
No it'd be eight times,
I believe.

00:02:01.480 --> 00:02:03.320
Let's see if eight
times works.

00:02:03.320 --> 00:02:04.850
There's always a little
bit of an art to this.

00:02:04.850 --> 00:02:06.900
If this is too little, we might
have to increase it.

00:02:06.900 --> 00:02:11.860
8 times 4 is 32.

00:02:11.860 --> 00:02:16.280
8 times 2 is 16 plus 3 is 19.

00:02:16.280 --> 00:02:17.320
No, that was right.

00:02:17.320 --> 00:02:18.520
Eight times.

00:02:18.520 --> 00:02:21.890
200 minus 192, that's 8.

00:02:21.890 --> 00:02:23.370
That is 8.

00:02:23.370 --> 00:02:24.620
Bring down another 0.

00:02:30.250 --> 00:02:33.690
24 goes into 80 three times.

00:02:33.690 --> 00:02:35.850
Can't be four times.

00:02:35.850 --> 00:02:38.890
3 times 4 is 12.

00:02:38.890 --> 00:02:39.600
Carry the 1.

00:02:39.600 --> 00:02:41.510
3 times 2 is 6.

00:02:41.510 --> 00:02:46.060
Add 170 and then you
have another 80.

00:02:46.060 --> 00:02:47.400
You bring down another 0.

00:02:47.400 --> 00:02:50.780
We have infinite supply of
0's here as we need them.

00:02:50.780 --> 00:02:53.100
And we're going to
have another 80.

00:02:53.100 --> 00:02:55.670
Well once again, 24 is going to
go into that three times.

00:02:55.670 --> 00:02:58.020
And we're going to start
repeating these 3's over and

00:02:58.020 --> 00:02:59.240
over and over again.

00:02:59.240 --> 00:03:02.380
So 5 over 24-- let me
write this down.

00:03:02.380 --> 00:03:04.260
5 over 24-- I want
you to understand

00:03:04.260 --> 00:03:05.030
why it keeps repeating.

00:03:05.030 --> 00:03:07.250
Every time we do this now,
we're going to get a 3.

00:03:07.250 --> 00:03:09.110
And come down, get a 72.

00:03:09.110 --> 00:03:11.320
80 from 72, we're going to get
another 80, and we're just

00:03:11.320 --> 00:03:13.430
going to have a big string
of 3's there.

00:03:13.430 --> 00:03:22.330
So 5 divided by 24 is
0.2083 repeating.

00:03:22.330 --> 00:03:25.490
Now, if we want to write this
as a percentage, the word

00:03:25.490 --> 00:03:29.030
percentage-- the word percent.

00:03:29.030 --> 00:03:29.910
Let me split it up.

00:03:29.910 --> 00:03:31.990
Per cent.

00:03:31.990 --> 00:03:34.850
Cent comes from the
word for 100.

00:03:34.850 --> 00:03:36.720
This is per 100.

00:03:36.720 --> 00:03:37.530
So what is this?

00:03:37.530 --> 00:03:39.750
This is 0.20.

00:03:39.750 --> 00:03:43.210
What is this per 100, where
100 is a whole?

00:03:43.210 --> 00:03:45.060
And that might confuse
you or it might not.

00:03:45.060 --> 00:03:48.450
But one way you can do this is
this is the same thing as--

00:03:48.450 --> 00:03:50.490
you could do it this way.

00:03:50.490 --> 00:03:51.600
This is the same thing.

00:03:51.600 --> 00:04:01.500
This decimal 0.2083 repeating
is the same thing as 20.83

00:04:01.500 --> 00:04:04.460
repeating over 100.

00:04:04.460 --> 00:04:08.010
This is how many we
have per 100%.

00:04:08.010 --> 00:04:17.579
Or you could say that this is
equal to 20.83% per 100.

00:04:17.579 --> 00:04:18.630
These are equivalent.

00:04:18.630 --> 00:04:21.110
A very quick way to think about
how to go from decimals

00:04:21.110 --> 00:04:26.060
to percent is you can multiply
by 100, and then put the

00:04:26.060 --> 00:04:26.800
percentage sign.

00:04:26.800 --> 00:04:30.100
Or, if you go backwards, you
divide by 100 and get rid of

00:04:30.100 --> 00:04:31.600
the percentage sign.

00:04:31.600 --> 00:04:32.500
That was a good warm up.

00:04:32.500 --> 00:04:34.330
Let's do one more.

00:04:34.330 --> 00:04:38.182
Let's convert a percentage
into a fraction.

00:04:38.182 --> 00:04:39.800
I'll do it in blue.

00:04:39.800 --> 00:04:43.620
So let's say we have 16%.

00:04:43.620 --> 00:04:46.396
Remember this means
16 per 100.

00:04:46.396 --> 00:04:49.390
So this is the same thing
as 16 per 100.

00:04:49.390 --> 00:04:52.830
That's what per cent--
century is 100 years.

00:04:52.830 --> 00:04:55.160
There are 100 cents
in a dollar.

00:04:55.160 --> 00:04:56.730
So 16 over 100.

00:04:56.730 --> 00:04:58.290
Put that in the lowest-- let's
see, you can divide the

00:04:58.290 --> 00:05:00.500
numerator and the denominator
by 4.

00:05:00.500 --> 00:05:04.950
We get 4 over 25.

00:05:04.950 --> 00:05:06.070
So that's our warm up.

00:05:06.070 --> 00:05:10.220
Let's do some actual
problems now.

00:05:10.220 --> 00:05:10.770
All right.

00:05:10.770 --> 00:05:14.240
They tell us a TV is
advertised on sale.

00:05:14.240 --> 00:05:19.800
It is 35% off, and has
a new price of $195.

00:05:19.800 --> 00:05:22.630
What was the pre-sale price?

00:05:22.630 --> 00:05:31.790
So if x is the pre-sale price,
when you take 35% off of that,

00:05:31.790 --> 00:05:34.850
it has a new price of $195.

00:05:34.850 --> 00:05:43.260
So x minus 35% of x-- 0.35
is the same thing as 35%.

00:05:43.260 --> 00:05:49.150
So if I take x and I subtract
35% of x from x, I'm

00:05:49.150 --> 00:05:52.320
going to get $195.

00:05:52.320 --> 00:05:54.110
That's what that
is telling me.

00:05:54.110 --> 00:05:56.090
So now we just solve for x.

00:05:56.090 --> 00:06:02.750
So you can view this
as 1x minus 0.35x.

00:06:02.750 --> 00:06:06.250
That will be 0.65x.

00:06:06.250 --> 00:06:10.380
So this is 0.65x, right?

00:06:10.380 --> 00:06:13.770
If you add 0.65 and
0.35, you get 1.

00:06:13.770 --> 00:06:19.870
1 minus 0.35 is 0.65x
is equal to 195.

00:06:19.870 --> 00:06:24.080
And now you can divide both
sides of this equation.

00:06:24.080 --> 00:06:26.380
Actually, I like to do it more
as a fraction, so let me write

00:06:26.380 --> 00:06:28.050
it that way.

00:06:28.050 --> 00:06:35.050
So if you have 65 over 100x is
equal to 195, now we can

00:06:35.050 --> 00:06:38.080
multiply both sides of this
equation by the inverse.

00:06:38.080 --> 00:06:39.210
Actually, even before
I do that, let me

00:06:39.210 --> 00:06:40.810
simplify this fraction.

00:06:40.810 --> 00:06:44.400
I can divide the numerator
and denominator by 5.

00:06:44.400 --> 00:06:48.130
This becomes 13 over 20.

00:06:48.130 --> 00:06:53.520
So we get 13 over 20x
is equal to 195.

00:06:53.520 --> 00:06:56.430
And now we can multiply both
sides by the inverse of this.

00:06:56.430 --> 00:07:03.990
So 20 over 13 times that is
equal to 195 times 20 over 13.

00:07:03.990 --> 00:07:12.140
These cancel out, and we get x
is equal to 195 times 20 over

00:07:12.140 --> 00:07:14.600
13, which we could figure out,
but I'll take the calculator

00:07:14.600 --> 00:07:16.520
out for this one.

00:07:16.520 --> 00:07:16.850
So let's see.

00:07:16.850 --> 00:07:26.910
You get 195 times 20 divided
by 13 is equal to $300.

00:07:26.910 --> 00:07:30.910
I should have done that-- so
this is equal to-- that right

00:07:30.910 --> 00:07:33.940
there is equal to $300.

00:07:33.940 --> 00:07:36.340
So $195 is actually
divisible by 13.

00:07:36.340 --> 00:07:38.240
I should be able to do that
without a calculator.

00:07:38.240 --> 00:07:40.990
Anyway, so the original
price was $300.

00:07:40.990 --> 00:07:47.960
You take 35% of $300, which
is $105, and you will

00:07:47.960 --> 00:07:50.080
be left with $195.

00:07:50.080 --> 00:07:54.190
So that was our original
price.

00:07:54.190 --> 00:07:56.280
Let's do another one.

00:07:56.280 --> 00:08:02.240
An employee at a store is
currently paid $9.50 per hour.

00:08:02.240 --> 00:08:06.090
If she works a full year, she
gets a 12% pay raise.

00:08:06.090 --> 00:08:09.510
What will be her new hourly
rate after the raise?

00:08:09.510 --> 00:08:12.050
So today, she makes 9.50.

00:08:12.050 --> 00:08:18.050
If she works for 12 years,
she'll make 9.50 plus 12%.

00:08:18.050 --> 00:08:23.170
Let me write this way;
plus 12% times 9.50.

00:08:23.170 --> 00:08:24.460
That's how much her
raise will be.

00:08:24.460 --> 00:08:27.320
So her new hourly rate will
be this entire thing.

00:08:27.320 --> 00:08:28.580
Well, we could view
it this way.

00:08:28.580 --> 00:08:40.049
This is the same thing as 9.50
plus 0.12 times 9.50.

00:08:40.049 --> 00:08:42.580
This is the same thing
as 1 times 9.50

00:08:42.580 --> 00:08:45.580
times 0.12 times 9.50.

00:08:45.580 --> 00:08:48.140
So this is the same
thing as 1.12.

00:08:48.140 --> 00:08:49.430
I'm just adding that.

00:08:49.430 --> 00:08:50.630
Let me do this in a
different color.

00:08:50.630 --> 00:08:59.630
I'm just adding that and that
to get that, times 9.50.

00:08:59.630 --> 00:09:02.570
We are growing 9.50 by 12%.

00:09:02.570 --> 00:09:05.370
You already have the 9.50
plus another 12%.

00:09:05.370 --> 00:09:10.470
It's 1.12 times 9.50 is going to
be equal to our new number.

00:09:10.470 --> 00:09:13.580
Let me multiply this just to
make up for me using the

00:09:13.580 --> 00:09:14.640
calculator the last time.

00:09:14.640 --> 00:09:20.550
So 9.50 times 1.12.

00:09:20.550 --> 00:09:22.350
2 times 0 is 0.

00:09:22.350 --> 00:09:25.430
2 times 5 is 10.

00:09:25.430 --> 00:09:30.520
2 times 9 is 18 plus 1 is 19.

00:09:30.520 --> 00:09:33.540
And I'll put a 0 here.

00:09:33.540 --> 00:09:35.740
1 times 0 is 0.

00:09:35.740 --> 00:09:37.320
1 times 5 is 5.

00:09:37.320 --> 00:09:39.340
1 times 9 is 9.

00:09:39.340 --> 00:09:42.020
Let's put two 0's here.

00:09:42.020 --> 00:09:44.200
1 times 0 is 0.

00:09:44.200 --> 00:09:46.220
1 times 5 is 5.

00:09:46.220 --> 00:09:48.260
1 times 9 is 9.

00:09:48.260 --> 00:09:51.510
And now we can add.

00:09:51.510 --> 00:09:54.990
The 0's add up to 0.

00:09:54.990 --> 00:09:58.420
9 plus 5 is 14.

00:09:58.420 --> 00:10:06.360
2 plus 9 is 11 plus 5 is 16.

00:10:06.360 --> 00:10:08.890
1 plus 9 is 10.

00:10:08.890 --> 00:10:10.970
And then we count the
number of decimals.

00:10:10.970 --> 00:10:15.120
We have one, two, three, four
numbers behind the decimals.

00:10:15.120 --> 00:10:19.000
So in our answer, we have to
have one, two, three, four

00:10:19.000 --> 00:10:21.640
numbers behind the decimal.

00:10:21.640 --> 00:10:27.110
So her new salary will
be $10.64 per hour.

00:10:27.110 --> 00:10:28.530
$10.64.

00:10:28.530 --> 00:10:31.770
You don't have to write those
trailing 0's right there.

00:10:31.770 --> 00:10:36.970
So this is her new hourly
rate after the raise.

00:10:36.970 --> 00:10:37.360
All right.

00:10:37.360 --> 00:10:38.610
Let's do one more.

00:10:41.130 --> 00:10:43.630
Store A and store B
both sell bikes.

00:10:43.630 --> 00:10:45.090
And both buy bikes
from the same

00:10:45.090 --> 00:10:46.950
supplier at the same price.

00:10:46.950 --> 00:10:51.440
Store A has a 40% mark-up.

00:10:51.440 --> 00:10:53.820
That means whatever they buy the
bike for, they sell it for

00:10:53.820 --> 00:10:56.200
40% above that.

00:10:56.200 --> 00:11:03.070
While store B has
a 250% markup.

00:11:03.070 --> 00:11:06.940
Store B has a permanent sale,
and always will sell at 60%

00:11:06.940 --> 00:11:08.400
off those prices.

00:11:08.400 --> 00:11:09.880
Which store has a better deal?

00:11:09.880 --> 00:11:13.470
So let's say that they're both
buying the bikes, so x is

00:11:13.470 --> 00:11:15.443
equal to the price
from supplier.

00:11:19.860 --> 00:11:22.850
So that's the price that
both bike stores buy

00:11:22.850 --> 00:11:24.070
their bicycles at.

00:11:24.070 --> 00:11:26.980
They both buy bikes from the
supplier at the same price.

00:11:26.980 --> 00:11:30.420
That's this x that I'm going
to start off with.

00:11:30.420 --> 00:11:33.120
Now lets do the scenario
of store A.

00:11:33.120 --> 00:11:35.700
What does store A sell
their bike for?

00:11:35.700 --> 00:11:43.930
They sell the bike for x plus
40% of x, which is equal to

00:11:43.930 --> 00:11:46.690
1.4 times x.

00:11:46.690 --> 00:11:49.720
That's how much store A
sells their bike for.

00:11:49.720 --> 00:11:52.810
Now store B is a little
bit more interesting.

00:11:52.810 --> 00:11:58.320
Store B-- so they have
a 250% markup.

00:11:58.320 --> 00:12:03.070
Well, they claim to sell their
bike for x plus 250%.

00:12:03.070 --> 00:12:08.020
That's the same thing
as 2.5 times x.

00:12:08.020 --> 00:12:11.450
A 100% markup would literally
mean another x.

00:12:11.450 --> 00:12:12.785
This is 250%.

00:12:12.785 --> 00:12:16.280
And as we said, you can
just divide by 100.

00:12:16.280 --> 00:12:17.400
You get 2.5.

00:12:17.400 --> 00:12:18.930
Get rid of the percentage
sign.

00:12:18.930 --> 00:12:24.780
So they sell it x plus 2.5x,
which is equal to 3.5x.

00:12:24.780 --> 00:12:26.620
That's kind of their
ticket price.

00:12:26.620 --> 00:12:30.380
But then, they sell at
60% off those prices.

00:12:30.380 --> 00:12:34.290
So then they take 60% off of
this price right here.

00:12:34.290 --> 00:12:40.200
So the real selling price is
going to be 3.5 times the

00:12:40.200 --> 00:12:44.560
price from the supplier
minus 0.6.

00:12:44.560 --> 00:12:46.250
Minus 0.6.

00:12:46.250 --> 00:12:49.450
Minus 60% of this price.

00:12:49.450 --> 00:12:50.700
Of 3.5x.

00:12:52.780 --> 00:12:56.860
And we could view this
as 1 times 3.5x

00:12:56.860 --> 00:12:59.480
minus 0.6 times 3.5X.

00:12:59.480 --> 00:13:09.300
And that's the same thing as
1 minus 0.6 times 3.5x.

00:13:09.300 --> 00:13:12.590
This is going to be 0.4.

00:13:12.590 --> 00:13:14.185
This is 0.4.

00:13:14.185 --> 00:13:17.390
So if you're taking 60% off,
it's the equivalent of selling

00:13:17.390 --> 00:13:19.780
it at 40% of the ticket price.

00:13:19.780 --> 00:13:20.790
That's what we're doing
right there.

00:13:20.790 --> 00:13:22.660
You might, eventually, do this
in your head-- immediately

00:13:22.660 --> 00:13:26.500
say, oh, 60% off is the same
thing as selling at 40%, or

00:13:26.500 --> 00:13:29.300
selling at 0.4 of the price.

00:13:29.300 --> 00:13:30.250
3.5x.

00:13:30.250 --> 00:13:33.320
And let's multiply that out.

00:13:33.320 --> 00:13:41.390
So if we take 3.5x times
0.4, 4 times 5 is 20.

00:13:41.390 --> 00:13:44.882
4 times 3 is 12 plus 2 is 14.

00:13:44.882 --> 00:13:50.230
And we have one, two numbers
behind the decimal spot.

00:13:50.230 --> 00:13:50.940
Two numbers.

00:13:50.940 --> 00:13:52.070
One, two.

00:13:52.070 --> 00:13:53.890
1.4.

00:13:53.890 --> 00:13:55.555
So they're going to
sell it at 1.4x.

00:13:58.460 --> 00:14:00.620
So in either store, you
actually going to

00:14:00.620 --> 00:14:02.000
pay the same price.

00:14:02.000 --> 00:14:05.450
Let's say they buy the bikes
from the supplier at $100.

00:14:05.450 --> 00:14:08.180
Then at store A, you're going to
pay $140 for that bike, and

00:14:08.180 --> 00:14:10.980
in store B, you're going to
spend $140 for that bike.

00:14:10.980 --> 00:14:14.360
But in store B, you think
you're getting 60% off.