Solving systems of inequalities word problem

WEBVTT
Kind: captions
Language: en

00:00:00.206 --> 00:00:03.292
- [Voiceover] "Fleur wants
to make tables and chairs.

00:00:03.292 --> 00:00:06.860
"Each chair or table is
made with the same number

00:00:06.860 --> 00:00:08.993
"of wooden boards and nails.

00:00:08.993 --> 00:00:12.223
"She has a total of 150 wooden boards

00:00:12.223 --> 00:00:14.853
and 330 nails.

00:00:14.853 --> 00:00:17.045
"The following inequality
represents the number

00:00:17.045 --> 00:00:19.589
"of tables, T, and chairs, C,

00:00:19.589 --> 00:00:23.246
"she can make with 150 wooden boards."

00:00:23.246 --> 00:00:26.566
So we can see the 150 wooden
boards right over here

00:00:26.566 --> 00:00:30.919
and it looks like she uses 17
wooden boards for each table

00:00:30.919 --> 00:00:34.467
so the total amount of
boards from the tables are

00:00:34.467 --> 00:00:37.428
to make the tables is 17 T.

00:00:37.428 --> 00:00:41.352
And it looks like she
uses six wooden boards

00:00:41.352 --> 00:00:43.348
for each chair and that's why you take

00:00:43.348 --> 00:00:45.368
six per chair times the number of chairs.

00:00:45.368 --> 00:00:47.110
This is the total number of wooden boards

00:00:47.110 --> 00:00:48.468
she uses from the chairs.

00:00:48.468 --> 00:00:49.373
And you add them together.

00:00:49.373 --> 00:00:50.987
It can't be anymore than 150.

00:00:50.987 --> 00:00:53.634
She only has, I guess 150 wooden boards.

00:00:53.634 --> 00:00:56.193
"Additionally, the following
inequality represents

00:00:56.193 --> 00:00:57.593
"the number of tables and chairs

00:00:57.593 --> 00:01:00.170
"she can make with 330 nails."

00:01:00.170 --> 00:01:03.769
So once again, she has
the maximum of 330 nails.

00:01:03.769 --> 00:01:06.404
It looks like she needs 34 nails per table

00:01:06.404 --> 00:01:09.718
because the total nails from the tables is

00:01:09.718 --> 00:01:13.129
34 nails per table times
the number of tables.

00:01:13.129 --> 00:01:16.329
And then it looks like she needs 27 nails.

00:01:16.329 --> 00:01:20.869
It looks like she needs 27 nails per chair

00:01:20.869 --> 00:01:22.563
because the total nails that she uses

00:01:22.563 --> 00:01:25.180
in all of the chairs
are 27 nails per chair

00:01:25.180 --> 00:01:27.205
times the number of chairs.

00:01:27.205 --> 00:01:28.714
"Does Fleur have enough boards

00:01:28.714 --> 00:01:31.755
"and nails to make three tables,

00:01:31.755 --> 00:01:33.695
"to make three,

00:01:34.135 --> 00:01:38.175
"to make three tables and nine chairs?

00:01:38.175 --> 00:01:40.253
"And nine chairs?"

00:01:40.253 --> 00:01:43.167
Let's see, let's look
at the number of boards.

00:01:43.167 --> 00:01:45.013
To make three tables and nine chairs,

00:01:45.013 --> 00:01:48.414
she's going to use 17 times T.

00:01:48.414 --> 00:01:49.796
In this case, it's three.

00:01:49.796 --> 00:01:54.796
Shes' going to use 17 times
three boards for the tables

00:01:54.950 --> 00:01:57.713
and then she is going to use,

00:01:57.713 --> 00:01:59.048
she is going to use

00:01:59.048 --> 00:02:02.914
six times the number of chairs.

00:02:02.914 --> 00:02:04.551
So six times nine chairs.

00:02:04.551 --> 00:02:07.047
Six boards per chair times nine chairs.

00:02:07.047 --> 00:02:08.614
So six times nine is the number of boards

00:02:08.614 --> 00:02:10.228
she's going to use on the chairs.

00:02:10.228 --> 00:02:12.085
So this is going to,
this is the same thing

00:02:12.085 --> 00:02:16.868
as 17 times three is 51.

00:02:16.868 --> 00:02:19.399
51 plus,

00:02:19.399 --> 00:02:22.199
six times nine is 54.

00:02:22.419 --> 00:02:24.472
So this is 54.

00:02:24.472 --> 00:02:25.412
And what is this?

00:02:25.412 --> 00:02:28.367
Is this less than or equal to 150?

00:02:28.367 --> 00:02:30.524
Is that less than or equal to 150?

00:02:30.524 --> 00:02:34.003
Let's see, 51 plus 54 is going to be 105.

00:02:34.003 --> 00:02:36.824
So that indeed is less
than or equal to 150.

00:02:36.824 --> 00:02:38.104
So that checks out.

00:02:38.104 --> 00:02:40.753
So she has more than enough boards.

00:02:40.753 --> 00:02:42.678
So she has enough boards.

00:02:42.678 --> 00:02:43.905
Let me write that.

00:02:43.905 --> 00:02:45.940
Enough boards.

00:02:45.940 --> 00:02:48.772
Now let's see if she has enough nails.

00:02:48.772 --> 00:02:49.835
Let's see if she has,

00:02:49.835 --> 00:02:50.992
that came out weird,

00:02:50.992 --> 00:02:52.511
enough boards.

00:02:52.511 --> 00:02:54.240
Let's see if she has enough nails.

00:02:54.240 --> 00:02:56.230
So three tables are going to require

00:02:56.230 --> 00:03:00.110
34 times three nails.

00:03:00.110 --> 00:03:02.969
So 34 nails per table times three tables.

00:03:02.969 --> 00:03:05.152
So 34 times three.

00:03:05.152 --> 00:03:07.620
That's how many nails
for all of the tables.

00:03:07.620 --> 00:03:10.767
And then plus,

00:03:10.767 --> 00:03:13.508
if she is going to use 27 nails per chair

00:03:13.508 --> 00:03:14.594
times the nine chairs.

00:03:14.594 --> 00:03:17.547
27 times nine.

00:03:17.877 --> 00:03:18.750
We need to figure out,

00:03:18.750 --> 00:03:23.070
is this less than or equal to 330?

00:03:23.070 --> 00:03:24.011
And so let's see.

00:03:24.011 --> 00:03:28.457
34 times three, that is 100.

00:03:28.457 --> 00:03:30.930
No, that's 90 plus 12.

00:03:30.930 --> 00:03:32.729
So that's 102

00:03:32.729 --> 00:03:36.954
and then 27 times nine.

00:03:36.954 --> 00:03:38.313
So let's see, that would be,

00:03:38.313 --> 00:03:41.134
was it 243?

00:03:41.134 --> 00:03:44.899
Yep, so plus 243.

00:03:44.899 --> 00:03:47.226
So it's 102 plus 243,

00:03:47.226 --> 00:03:50.733
is that less than or equal to 330?

00:03:50.733 --> 00:03:52.379
So let's see, this would be,

00:03:52.379 --> 00:03:56.050
this would end up adding up to 345.

00:03:56.050 --> 00:04:01.050
No, this is not less than or equal to 330.

00:04:01.146 --> 00:04:02.284
This is not the case.

00:04:02.284 --> 00:04:03.933
This is not true.

00:04:03.933 --> 00:04:06.545
Not true.

00:04:06.545 --> 00:04:09.029
So she does not have enough nails.

00:04:09.029 --> 00:04:11.722
So not enough,

00:04:12.152 --> 00:04:14.692
not enough nails.

00:04:15.693 --> 00:04:18.363
And so she has enough
boards but not enough nails.

00:04:18.363 --> 00:04:20.196
So unless she goes and buys some nails,

00:04:20.196 --> 00:04:23.431
she's not going to be able to
make these tables and chairs.


Writing systems of inequalities word problem

WEBVTT
Kind: captions
Language: en

00:00:00.589 --> 00:00:03.162
- [Voiceover] So we're told
a cupcake requires 35 grams

00:00:03.162 --> 00:00:05.623
of sugar and 50 grams of flour.

00:00:05.623 --> 00:00:07.780
So let me underline that.

00:00:07.780 --> 00:00:10.042
"A cupcake requires 35 grams of sugar

00:00:10.042 --> 00:00:12.306
"and 50 grams of flour."

00:00:12.306 --> 00:00:15.661
And then they tell us,

00:00:15.661 --> 00:00:18.888
"A muffin requires 30 grams of sugar."

00:00:18.888 --> 00:00:19.810
So let me underline that.

00:00:19.810 --> 00:00:22.452
"A muffin requires 30 grams of sugar

00:00:22.452 --> 00:00:25.342
and 65 grams of flour."

00:00:25.342 --> 00:00:28.768
All right, so what are they
going to ask us to actually do?

00:00:28.768 --> 00:00:33.768
"Shawna needs to use up at
least 460 grams of sugar."

00:00:33.814 --> 00:00:35.617
So let me underline
this in different color.

00:00:35.617 --> 00:00:37.791
She needs to use up,

00:00:38.011 --> 00:00:42.780
"Shawna needs to use up at
least 460 grams of sugar

00:00:42.780 --> 00:00:45.543
"to make cupcakes and muffins,

00:00:45.543 --> 00:00:50.360
"and she wants to use at
most 970 grams of flour."

00:00:50.360 --> 00:00:52.137
So let me use another color.

00:00:52.137 --> 00:00:54.375
"And she wants to use

00:00:54.375 --> 00:00:57.955
"at most 970 grams of flour.

00:00:57.955 --> 00:00:59.984
"Let's form a system of inequalities

00:00:59.984 --> 00:01:02.399
"to represent Shawna's conditions.

00:01:02.399 --> 00:01:05.545
"Let C denote the number
of cupcakes she makes

00:01:05.545 --> 00:01:08.215
"and M the number of muffins she makes."

00:01:08.215 --> 00:01:11.535
All right, "Write an
inequality that represents

00:01:11.535 --> 00:01:16.376
"the condition based on the
number of grams of sugar."

00:01:16.376 --> 00:01:18.837
So the condition based on
the number of grams of sugar.

00:01:18.837 --> 00:01:20.822
That's what we underlined here in green.

00:01:20.822 --> 00:01:24.758
"Shawna needs to use up at
least 460 grams of sugar

00:01:24.758 --> 00:01:27.138
"to make cupcakes and muffins."

00:01:27.138 --> 00:01:28.635
So let's see how we can write that out.

00:01:28.635 --> 00:01:32.199
So how much sugar she's
gonna use for the cupcakes?

00:01:32.199 --> 00:01:34.835
Well, we're told right over here

00:01:34.835 --> 00:01:38.480
she requires 35 grams
of sugar per cupcake.

00:01:38.480 --> 00:01:40.012
So the amount of sugar
that she's going to use

00:01:40.012 --> 00:01:41.819
for all of the cupcakes is going to be

00:01:41.819 --> 00:01:45.469
35 grams per cupcake,

00:01:45.469 --> 00:01:46.919
let me write that a little bit neater,

00:01:46.919 --> 00:01:50.994
35 grams per cupcake times
the number of cupcakes.

00:01:50.994 --> 00:01:52.535
So this is how much sugar total

00:01:52.535 --> 00:01:53.920
she's going to use on the cupcakes.

00:01:53.920 --> 00:01:55.777
35 grams per cupcake

00:01:55.777 --> 00:01:57.519
times the number of cupcakes.

00:01:57.519 --> 00:01:58.830
And then how many grams of sugar is

00:01:58.830 --> 00:02:01.829
she going to use for the muffins?

00:02:01.829 --> 00:02:05.459
Well, that's going to be 30
grams of sugar per muffin.

00:02:05.459 --> 00:02:08.502
So plus 30 grams

00:02:08.502 --> 00:02:10.380
of sugar per muffin

00:02:10.380 --> 00:02:12.516
times the number of muffins.

00:02:12.516 --> 00:02:14.281
So times M.

00:02:14.281 --> 00:02:16.336
So this is the total
sugar from the cupcakes.

00:02:16.336 --> 00:02:18.437
This is the total sugar from the muffins.

00:02:18.437 --> 00:02:19.725
And so if you add them together,

00:02:19.725 --> 00:02:22.036
this is a total sugar that she's using

00:02:22.036 --> 00:02:24.241
and we're told that she needs to use up

00:02:24.241 --> 00:02:26.842
at least 460 grams of sugar.

00:02:26.842 --> 00:02:28.641
So everything here is in grams.

00:02:28.641 --> 00:02:31.323
So this has to be greater than or equal.

00:02:31.323 --> 00:02:35.723
She needs to use at
least 460 grams of sugar

00:02:35.723 --> 00:02:37.243
to make both of these.

00:02:37.243 --> 00:02:37.765
So that's it.

00:02:37.765 --> 00:02:39.996
We can just set up the
inequality that represents

00:02:39.996 --> 00:02:42.700
the condition based on the
number of grams of sugar

00:02:42.700 --> 00:02:45.637
which is this condition right over here.

00:02:45.637 --> 00:02:47.297
Now let's see, this next question.

00:02:47.297 --> 00:02:50.280
Let's see, "Write an
inequality that represents

00:02:50.280 --> 00:02:53.923
"the condition based on the
number of grams of flour."

00:02:53.923 --> 00:02:56.345
Well, that's what we underlined
in orange right over here.

00:02:56.345 --> 00:03:00.274
She needs to use no at
most 970 grams of flour.

00:03:00.274 --> 00:03:02.597
So no more than 970 grams.

00:03:02.597 --> 00:03:05.697
So how would we express
that as an inequality?

00:03:05.697 --> 00:03:08.532
Well, how much flour is she
going to use for the cupcakes?

00:03:08.532 --> 00:03:11.312
Well, she used 50 grams
of flour per cupcake.

00:03:11.312 --> 00:03:15.274
So 50 grams per cupcake
times the number of cupcakes.

00:03:15.274 --> 00:03:17.933
That's how much flour she's
going to use for the cupcakes.

00:03:17.933 --> 00:03:19.779
And then for the muffins,

00:03:20.139 --> 00:03:24.068
she uses 65 grams of flour per muffin

00:03:24.538 --> 00:03:27.452
times the number of muffins.

00:03:27.452 --> 00:03:29.332
So this is the total amount of flour

00:03:29.332 --> 00:03:31.214
that she is going to use

00:03:31.214 --> 00:03:36.214
and this has to be less
than or equal to 970 grams.

00:03:36.519 --> 00:03:40.373
Less than or equal to 970

00:03:40.373 --> 00:03:41.615
and we're done.

00:03:41.615 --> 00:03:43.652
And so you could actually,
if you wanted to,

00:03:43.652 --> 00:03:47.694
you could say, "What pairs of C and M

00:03:47.694 --> 00:03:50.357
"satisfy both of these inequalities?"

00:03:50.357 --> 00:03:52.493
And then you could say,
"Okay, that's the combination

00:03:52.493 --> 00:03:55.035
"of the number of cupcakes
and muffins she can make

00:03:55.035 --> 00:03:58.275
"in order to satisfy
her actual conditions."


Graphs of two-variable inequalities word problem

WEBVTT
Kind: captions
Language: en

00:00:00.665 --> 00:00:02.859
- [Voiceover] Diana the dog receives five

00:00:02.859 --> 00:00:05.582
dog biscuits for fetching each frisbee,

00:00:05.582 --> 00:00:08.264
and three dog biscuits
for fetching each ball.

00:00:08.264 --> 00:00:10.072
Sounds like a pretty good deal.

00:00:10.072 --> 00:00:12.144
She plans to receive at most D

00:00:12.144 --> 00:00:14.765
dog biscuits before chasing her tail.

00:00:14.765 --> 00:00:16.248
Well, that's sounds reasonable.

00:00:16.248 --> 00:00:18.544
The inequality graph
below represents the set

00:00:18.544 --> 00:00:22.323
of all combinations where
Diana fetches F frisbees

00:00:22.323 --> 00:00:25.022
and fetches B balls in order to receive

00:00:25.022 --> 00:00:27.772
at most, at most, D dog biscuits

00:00:27.772 --> 00:00:30.148
'cause at that point she
reasonably starts chasing her tail.

00:00:30.148 --> 00:00:31.876
According to the graph, we're gonna take

00:00:31.876 --> 00:00:33.359
a look at the graph in a second,

00:00:33.359 --> 00:00:36.854
according to the graph, what
is the most number of dog

00:00:36.854 --> 00:00:40.714
biscuits Diana wants to receive
before chasing her tail?

00:00:40.714 --> 00:00:43.254
In other words, what is D?

00:00:43.254 --> 00:00:46.117
So let's look at, let's
interpret this graph properly.

00:00:46.832 --> 00:00:49.412
So if we look at the horizontal
axis right over here,

00:00:49.412 --> 00:00:52.199
that's F, that's the number of frisbees,

00:00:52.199 --> 00:00:57.177
the number of frisbees
she catches, frisbees

00:00:57.177 --> 00:01:00.671
and this vertical axis,
this is the number of balls,

00:01:00.671 --> 00:01:05.671
number of, number of balls that she gets

00:01:05.934 --> 00:01:07.863
and we know what the total number

00:01:07.863 --> 00:01:09.773
of biscuits are going to be.

00:01:09.773 --> 00:01:13.329
The total from catching frisbees,
if she catches F frisbees,

00:01:13.329 --> 00:01:15.767
she gets five biscuits per frisbee

00:01:15.767 --> 00:01:18.875
so the total from catching frisbees is 5F

00:01:18.875 --> 00:01:22.025
and if she catches, if she catches B balls

00:01:22.025 --> 00:01:26.575
or retrieves B balls so
if she gets those B balls,

00:01:26.575 --> 00:01:28.628
if she gets, what was it? Three?

00:01:28.628 --> 00:01:30.273
Three biscuits per ball? Yep!

00:01:30.273 --> 00:01:32.793
Three dog biscuits for fetching each ball,

00:01:32.793 --> 00:01:34.804
the total number of dog biscuits she gets

00:01:34.804 --> 00:01:39.804
for catching B balls is
3B and so the total number

00:01:40.505 --> 00:01:45.259
of biscuits she fetches is 5F plus 3B.

00:01:45.259 --> 00:01:46.925
This is the number of
biscuits from frisbees.

00:01:46.925 --> 00:01:49.444
This is the number of biscuits from balls.

00:01:49.444 --> 00:01:51.821
Now we can see all of the
allowable combinations

00:01:51.821 --> 00:01:54.950
of number of frisbees
and number of balls here.

00:01:54.950 --> 00:01:57.002
And so for example, if she catches

00:01:57.002 --> 00:01:58.383
that point right over there, eight

00:01:58.383 --> 00:02:00.943
or retrieves eight balls and catches,

00:02:00.943 --> 00:02:02.569
well, that would be half of a frisbee,

00:02:02.569 --> 00:02:05.108
so that doesn't, that
doesn't seem to make sense

00:02:05.108 --> 00:02:10.108
but if she retrieves eight
balls and catches one frisbee

00:02:10.797 --> 00:02:13.807
well, then, that's still
ok, she still hasn't met her

00:02:13.807 --> 00:02:16.500
maximum number of biscuits yet.

00:02:16.500 --> 00:02:18.675
So how do we think about the
maximum number of biscuits?

00:02:18.675 --> 00:02:20.827
Well, the maximum number of biscuits

00:02:20.827 --> 00:02:22.596
are any of these points when she,

00:02:22.596 --> 00:02:25.500
that are sitting on this line
and notice, the solutions

00:02:25.500 --> 00:02:27.533
that where all the points
it satisfies this inequality

00:02:27.533 --> 00:02:30.479
are all below this line
so she's hitting a maximum

00:02:30.479 --> 00:02:32.511
when she's on the line and an easy one

00:02:32.511 --> 00:02:36.798
might be this point right
over here where we see that,

00:02:36.798 --> 00:02:41.288
where we see that frisbees,
zero frisbees and 10

00:02:41.288 --> 00:02:45.818
and 10 balls pretty much
maximizes her number of biscuits.

00:02:45.818 --> 00:02:48.846
So if she catches 10 balls,
so let me write this down,

00:02:48.846 --> 00:02:51.934
so if B is equal to 10, B is equal to 10,

00:02:51.934 --> 00:02:56.126
F is zero, if F is equal to zero

00:02:56.126 --> 00:02:59.458
and B is equal to 10, well,
how many is she going to catch?

00:02:59.458 --> 00:03:02.201
Or how many biscuits is she going to get?

00:03:02.201 --> 00:03:04.562
Well, she's going to get,
this is gonna be zero

00:03:04.562 --> 00:03:07.182
and then three times 10 is 30,

00:03:07.182 --> 00:03:11.185
so that's gonna be 30
biscuits, 30 biscuits.

00:03:11.185 --> 00:03:13.764
So this point right over
here, this corresponds

00:03:13.764 --> 00:03:18.032
with 30 biscuits, 30 biscuits

00:03:19.693 --> 00:03:21.500
and you can see that any of these points

00:03:21.500 --> 00:03:24.828
along this blue line, actually
correspond to 30 biscuits.

00:03:24.828 --> 00:03:27.683
If you go over here, where F is six,

00:03:27.683 --> 00:03:30.507
let me write it here, F is equal to six

00:03:30.507 --> 00:03:33.940
and B is equal to zero so
spends all of her time,

00:03:33.940 --> 00:03:35.444
she's earns her biscuits purely

00:03:35.444 --> 00:03:38.268
through frisbee catching,
so this is a situation

00:03:38.268 --> 00:03:41.072
where F is equal to six
and B is equal to zero,

00:03:41.072 --> 00:03:42.819
you still have the same scenario,

00:03:42.819 --> 00:03:46.292
you still have, if F is
six, five times six is 30

00:03:46.292 --> 00:03:48.386
plus three times zero,
we'll that's just gonna be

00:03:48.386 --> 00:03:52.652
once again, 30 biscuits, 30 biscuits.

00:03:52.652 --> 00:03:55.111
So her maximum, or the number of biscuits

00:03:55.111 --> 00:03:58.482
she needs before starts
chasing her tail is 30.

00:03:58.482 --> 00:04:02.262
So D is going to be 30 and
in fact, we can express

00:04:02.262 --> 00:04:05.720
this inequality as a 5F plus 3B has

00:04:05.720 --> 00:04:09.581
to be less than or equal to 30.

00:04:09.581 --> 00:04:11.835
Alright, then they ask
us another question.

00:04:11.835 --> 00:04:14.070
Can Diana fulfill her plan by fetching,

00:04:14.070 --> 00:04:17.537
can Diana fulfill her plan by fetching

00:04:17.537 --> 00:04:20.069
four frisbees and two balls?

00:04:20.069 --> 00:04:23.056
So let's see, four frisbees and two balls,

00:04:23.558 --> 00:04:28.558
this is right over here,
four frisbees and two balls.

00:04:28.720 --> 00:04:30.000
So it's not, we're not saying that she

00:04:30.000 --> 00:04:32.803
has to maximize, that she
has to get the 30 biscuits,

00:04:32.803 --> 00:04:36.054
she just cannot eat any
more than 30 biscuits.

00:04:36.054 --> 00:04:41.054
So it seems like she can
fulfill her plan, let me see.

00:04:41.065 --> 00:04:43.400
The inequality graphed below
represents all the combinations

00:04:43.400 --> 00:04:45.677
where Diana fetches F frisbees and B balls

00:04:45.677 --> 00:04:48.480
in order to receive at
most D dog biscuits.

00:04:48.480 --> 00:04:53.480
So let's see, is she fulfilling
her plan by fetching,

00:04:54.164 --> 00:04:57.949
well, yeah, I would say, yes.


Solving two-variable inequalities word problem

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:01.892
- Ezra enjoys gardening.

00:00:02.107 --> 00:00:04.661
Every sunflower plant he waters

00:00:04.661 --> 00:00:08.469
requires 0.7 liters of water,

00:00:08.469 --> 00:00:10.793
and every lily plant he waters

00:00:10.793 --> 00:00:13.510
requires 0.5 liters of water.

00:00:13.727 --> 00:00:18.022
Ezra has a total of 11 liters
of water for these plants.

00:00:18.313 --> 00:00:19.857
In the following inequality,

00:00:19.857 --> 00:00:22.270
S represents the number
of sunflower plants

00:00:22.270 --> 00:00:24.451
and L represents the number

00:00:24.451 --> 00:00:26.818
of lily plants Ezra can water.

00:00:27.098 --> 00:00:28.245
See if this makes sense.

00:00:28.308 --> 00:00:29.800
So it looks like...

00:00:29.852 --> 00:00:32.378
So the number of sunflower plants is S

00:00:32.627 --> 00:00:36.603
and he would have to use
7/10 of a liter per plant.

00:00:36.992 --> 00:00:39.076
So this first term right over here,

00:00:39.372 --> 00:00:40.795
this is how much water used

00:00:40.795 --> 00:00:43.318
to water the sunflower plants

00:00:43.508 --> 00:00:45.492
and then the second term is how much water

00:00:45.492 --> 00:00:47.544
used to water the lily plants

00:00:47.544 --> 00:00:51.184
cause he needs half a liter
for each of the L lily plants.

00:00:51.387 --> 00:00:53.355
And he has a total of 11 liters.

00:00:53.453 --> 00:00:54.928
So the sum right over here

00:00:54.928 --> 00:00:56.675
has to be less than or equal to 11.

00:00:56.994 --> 00:00:57.639
Alright.

00:00:57.783 --> 00:01:00.495
Ezra waters 10 lily plants.

00:01:00.686 --> 00:01:02.561
So that's L is equal to 10.

00:01:02.903 --> 00:01:05.991
How many sunflower plants
can he water at most

00:01:05.991 --> 00:01:08.342
with the remaining amount of water?

00:01:08.626 --> 00:01:10.060
Well let's think about this a little bit.

00:01:10.333 --> 00:01:12.243
We have 0.7,

00:01:13.026 --> 00:01:16.573
0.7 times the number of
sunflower plants he waters.

00:01:17.043 --> 00:01:19.255
Plus, plus...

00:01:19.655 --> 00:01:21.491
So we're gonna assume L is equal to 10.

00:01:21.491 --> 00:01:22.963
He waters 10 lily plants

00:01:22.963 --> 00:01:26.000
and each of them, he
gives half a liter to.

00:01:26.307 --> 00:01:27.602
So 0.5,

00:01:28.501 --> 00:01:32.373
0.5 x 10 is going to be 5.

00:01:32.808 --> 00:01:36.407
So it's gonna be five liters,
watering the 10 lily plants

00:01:36.407 --> 00:01:38.897
and this has to be less
than or equal to 11.

00:01:40.110 --> 00:01:44.179
So we can try to isolate
the S on the left hand side.

00:01:44.243 --> 00:01:46.530
So let's, let's do that.
The number of sunflowers.

00:01:46.530 --> 00:01:48.509
So we could first subtract
five from both sides.

00:01:49.327 --> 00:01:51.458
Subtract five from both sides.

00:01:51.707 --> 00:01:53.617
and we are going to be left with...

00:01:54.168 --> 00:01:56.920
We are going to be left
with on the left hand side,

00:01:57.800 --> 00:02:01.558
we're gonna just have 0.7 x S

00:02:02.028 --> 00:02:04.001
So this expression is
the total amount of water

00:02:04.001 --> 00:02:05.772
he's gonna spend on the sunflower plants

00:02:06.044 --> 00:02:08.286
has to be less than or equal to

00:02:08.286 --> 00:02:10.716
11- 5 is equal to six.

00:02:11.456 --> 00:02:14.931
And so now, we can
divide both sides by 0.7

00:02:15.784 --> 00:02:17.677
And we're gonna have to
change the inequality

00:02:17.677 --> 00:02:21.073
because we are diving by
a positive number, 0.7

00:02:21.484 --> 00:02:23.093
So we're gonna get S...

00:02:24.038 --> 00:02:26.777
S is less than or equal to

00:02:26.777 --> 00:02:29.588
6 ÷ 0.7

00:02:30.038 --> 00:02:31.648
or we could even write this as...

00:02:31.805 --> 00:02:33.808
we could write this as, well...

00:02:33.825 --> 00:02:37.395
This is 6 divided by 0.7

00:02:37.424 --> 00:02:38.730
Let me write everything as a fraction.

00:02:38.898 --> 00:02:43.455
This is the same thing as 6 ÷ 7/10

00:02:43.576 --> 00:02:48.576
which is equal to six times 10 over seven

00:02:48.974 --> 00:02:52.510
which is equal to 60/7

00:02:52.620 --> 00:02:55.168
which is equal to what is this?

00:02:55.174 --> 00:02:57.711
8 4/7

00:02:57.902 --> 00:03:02.273
so S has to be less than
eight and four sevenths.

00:03:02.476 --> 00:03:03.683
Now assuming that he's only,

00:03:03.683 --> 00:03:06.208
we're only thinking in
terms of whole plants...

00:03:06.214 --> 00:03:07.009
(mumbles)

00:03:07.049 --> 00:03:09.947
integer numbers but actually
whole numbers of plants

00:03:10.080 --> 00:03:13.319
and so, if S has to be less than 8 4/7,

00:03:13.319 --> 00:03:15.426
how many sunflower plants can he water

00:03:15.431 --> 00:03:17.307
at most with the
remaining amount of water?

00:03:17.602 --> 00:03:18.861
Well, we'll say that he had,

00:03:18.861 --> 00:03:20.593
most he can water eight plants.

00:03:20.833 --> 00:03:22.572
We're gonna assume that he can't water

00:03:22.572 --> 00:03:24.057
just a fraction of a plant.

00:03:24.057 --> 00:03:25.746
He either waters something or he doesn't.

00:03:25.972 --> 00:03:27.627
So he's gonna water eight,

00:03:27.795 --> 00:03:30.088
he is gonna water eight plants.

00:03:30.999 --> 00:03:32.961
Cause that's the largest whole number

00:03:32.961 --> 00:03:35.631
that is less than 8 4/7.

00:03:35.631 --> 00:03:36.316
That satisfies...

00:03:36.316 --> 00:03:39.480
That is less than or equal
to 8 4/7 I should say.


Interpreting two-variable inequalities word problem

WEBVTT
Kind: captions
Language: en

00:00:00.771 --> 00:00:03.487
- [Voiceover] "Goku wants
to get over 9000 comments

00:00:03.487 --> 00:00:05.008
"on his YouTube videos.

00:00:05.008 --> 00:00:06.831
"He gets to the same number of comments

00:00:06.831 --> 00:00:09.025
"for each cat video that he uploads,

00:00:09.025 --> 00:00:10.533
"and he gets the same number of comments

00:00:10.533 --> 00:00:12.589
"for each dog video that he uploads."

00:00:12.589 --> 00:00:14.040
That's somewhat amazing.

00:00:14.040 --> 00:00:17.627
The exact same number of
comments for each cat video

00:00:17.627 --> 00:00:20.808
and the exact same number
for each dog video.

00:00:20.808 --> 00:00:23.431
But we'll just assume
what they're telling us.

00:00:23.431 --> 00:00:26.651
"In the following inequality, C represents

00:00:26.651 --> 00:00:29.631
"the number of cat videos, C for cat,

00:00:29.631 --> 00:00:31.739
"and D represents the
number of dog videos,

00:00:31.739 --> 00:00:35.826
"D for dog, dog videos
that Goku should upload

00:00:35.826 --> 00:00:37.567
"to achieve his goal."

00:00:37.567 --> 00:00:40.551
So 750 times the number of cat videos

00:00:40.551 --> 00:00:43.891
plus 450 times the number of dog videos

00:00:43.891 --> 00:00:47.307
needs to be greater than 9000.

00:00:47.307 --> 00:00:48.665
"According to the inequality,

00:00:48.665 --> 00:00:51.800
"how many comments does each cat video get

00:00:51.800 --> 00:00:54.974
"and how many comments
does each dog video get?"

00:00:54.974 --> 00:00:57.604
So let's see if we understand
what's going over here,

00:00:57.604 --> 00:01:00.518
what's going on in this inequality
to answer their question.

00:01:00.518 --> 00:01:02.956
So he wants to get over 9000 comments

00:01:02.956 --> 00:01:04.082
and we see that right over here.

00:01:04.082 --> 00:01:07.623
He wants to get over 9000 comments.

00:01:07.623 --> 00:01:09.658
So this we could view as a number

00:01:09.658 --> 00:01:12.508
of comments that he actually gets.

00:01:12.968 --> 00:01:15.796
Since we have some number times C,

00:01:15.796 --> 00:01:17.430
we can assume that, well C,

00:01:17.430 --> 00:01:19.606
well we know that C is
a number of cat videos

00:01:19.606 --> 00:01:21.614
and so the number times C,

00:01:21.614 --> 00:01:22.791
that's the number of comments

00:01:22.791 --> 00:01:24.934
that he gets from cat videos.

00:01:24.934 --> 00:01:27.291
And on a per cat video bases,

00:01:27.291 --> 00:01:29.969
he's going to get 750 comments.

00:01:29.969 --> 00:01:30.805
Because if I say, "Hey, look.

00:01:30.805 --> 00:01:32.852
"I have C cat videos and on each of them,

00:01:32.852 --> 00:01:35.456
"I have exactly 750 comments."

00:01:35.456 --> 00:01:39.097
Well, I'm gonna have 750 times C comments

00:01:39.097 --> 00:01:41.651
on all of my cat videos combined.

00:01:41.651 --> 00:01:46.631
So each cat video gets 750 comments.

00:01:46.631 --> 00:01:49.093
And similarly, I can never say that,

00:01:49.093 --> 00:01:50.853
each dog video gets,

00:01:50.853 --> 00:01:54.758
well this is the total number
of comments from dog videos.

00:01:54.758 --> 00:01:56.035
You add all of these together.

00:01:56.035 --> 00:01:57.695
Your total number of comments w

00:01:57.695 --> 00:01:59.459
to be greater than 9000.

00:01:59.459 --> 00:02:02.972
Well, if the total from
dog videos is 450 D,

00:02:02.972 --> 00:02:07.163
then we see he's getting
450 comments per dog video.

00:02:07.163 --> 00:02:10.599
450 comments per video times
the number of dog videos

00:02:10.599 --> 00:02:13.757
gets the total comments from dog videos.

00:02:13.757 --> 00:02:18.262
So he gets 450 comments per dog video.

00:02:18.262 --> 00:02:22.278
"Can Goku achieve this goal
by uploading eight cat videos

00:02:22.278 --> 00:02:24.414
"and seven dog videos?"

00:02:24.414 --> 00:02:25.710
So let's think about this.

00:02:25.710 --> 00:02:28.721
Eight cat videos, seven dog videos.

00:02:28.721 --> 00:02:29.441
So let's see.

00:02:29.441 --> 00:02:33.095
If we have 750, let me
do those in same colors,

00:02:33.095 --> 00:02:37.910
if we have 750 times eight,

00:02:37.910 --> 00:02:39.500
times eight

00:02:39.500 --> 00:02:44.138
plus 450

00:02:44.138 --> 00:02:47.035
times seven, is this greater than,

00:02:47.035 --> 00:02:49.182
I'm writing the question
mark so we don't know,

00:02:49.182 --> 00:02:51.174
is this going to be greater

00:02:51.174 --> 00:02:54.744
than 9000?

00:02:54.744 --> 00:02:56.564
Well, we just have to evaluate this.

00:02:56.564 --> 00:02:58.438
What is this going to be?

00:02:58.438 --> 00:02:59.314
Let's see.

00:02:59.314 --> 00:03:04.314
700 times eight is 5600.

00:03:04.539 --> 00:03:06.211
It's going to be 5600

00:03:06.211 --> 00:03:10.228
plus another 50 times eight is 400.

00:03:10.228 --> 00:03:14.709
So 5600 plus 400 is 60 hundred or 6000.

00:03:14.709 --> 00:03:18.203
So this is 6000 right over here.

00:03:18.203 --> 00:03:20.711
So that would be, if he
has eight cat videos,

00:03:20.711 --> 00:03:23.845
he gets 750 comments per video

00:03:23.845 --> 00:03:25.777
so he's going to have a
total of 6000 comments

00:03:25.777 --> 00:03:28.525
on all of the eight cat videos combined.

00:03:28.525 --> 00:03:31.664
And then 450 times seven,

00:03:31.664 --> 00:03:34.142
let's see, 400 times seven is 2800

00:03:34.142 --> 00:03:37.940
plus another 30, plus another 350,

00:03:37.940 --> 00:03:39.497
that's going to be, what?

00:03:39.497 --> 00:03:43.455
3150, is that right?

00:03:43.455 --> 00:03:44.625
Let's see, that would be,

00:03:44.625 --> 00:03:45.483
let me show you to this right,

00:03:45.483 --> 00:03:49.308
2800 plus 400

00:03:49.308 --> 00:03:51.348
times seven is 2800.

00:03:51.348 --> 00:03:54.379
50 times seven is 350.

00:03:54.379 --> 00:03:56.507
Yep, I think that is right.

00:03:56.507 --> 00:03:59.630
It all, all works out, 3150.

00:03:59.630 --> 00:04:04.186
So 6000 plus 3150 is
indeed greater than 9000.

00:04:04.186 --> 00:04:06.108
It is indeed greater than 9000.

00:04:06.108 --> 00:04:08.302
This over here is 9150

00:04:08.302 --> 00:04:11.286
which is indeed greater than 9000.

00:04:11.286 --> 00:04:13.085
So yes, he can achieve his goal

00:04:13.085 --> 00:04:17.921
by uploading eight cat
videos and seven dog videos.


Writing two-variable inequalities word problem

WEBVTT
Kind: captions
Language: en

00:00:00.525 --> 00:00:03.984
- [Voiceover] Fabiano wants
to score at least 6.5 points

00:00:03.984 --> 00:00:06.887
in a major chess tournament.

00:00:06.887 --> 00:00:10.440
He scores one point for
each game that he wins,

00:00:10.440 --> 00:00:15.440
and he scores 0.5 points for
each game that ends in a draw.

00:00:15.478 --> 00:00:18.345
Write an inequality that
represents the number of games

00:00:18.345 --> 00:00:23.338
Fabiano should win and draw,
D, to achieve his goal.

00:00:23.338 --> 00:00:25.219
So I encourage you to
pause the video and see if

00:00:25.219 --> 00:00:25.904
you can do that.

00:00:25.904 --> 00:00:27.899
Write an inequality in
terms of the number of

00:00:27.899 --> 00:00:31.986
games won, so capital "W",
and the number of draws,

00:00:31.986 --> 00:00:35.527
capital "D", that represents
what he needs to do to actually

00:00:35.527 --> 00:00:37.977
achieve his goal.

00:00:37.977 --> 00:00:39.276
All right, let's work through it together.

00:00:39.276 --> 00:00:41.959
So how many points he is
going to get from winning?

00:00:41.959 --> 00:00:44.240
So if he wins "W",

00:00:44.240 --> 00:00:46.700
he's gonna win W games,

00:00:46.700 --> 00:00:48.935
and he gets one point for
each of them, so it's gonna be

00:00:48.935 --> 00:00:52.036
one point per game, times
the number of games.

00:00:52.036 --> 00:00:54.497
So one W I could just write as W.

00:00:54.497 --> 00:00:57.034
So this is the points from winning.

00:01:02.182 --> 00:01:03.436
From wins, I could say.

00:01:03.436 --> 00:01:06.175
And what are his points
gonna be from the draws?

00:01:06.175 --> 00:01:09.577
Well, from the draws, he's gonna have

00:01:09.577 --> 00:01:14.577
D draws, and he gets 0.5
points for each of them.

00:01:14.586 --> 00:01:17.849
zero point five times D, this
is going to be the points

00:01:17.849 --> 00:01:19.621
from the draws.

00:01:28.258 --> 00:01:29.582
Now, this right over here
is going to be his total

00:01:29.582 --> 00:01:31.996
points, points from
wins, points from draws.

00:01:31.996 --> 00:01:34.441
I'm assuming he gets no points for losses.

00:01:34.441 --> 00:01:36.264
And we want this number, the total number

00:01:36.264 --> 00:01:40.536
of points, his score, to be at least 6.5.

00:01:40.536 --> 00:01:44.780
So we want this to be
greater than or equal to 6.5.

00:01:49.058 --> 00:01:52.587
If it says Fabiano wants
to score more than 6.5,

00:01:52.587 --> 00:01:53.817
then it would have been greater than.

00:01:53.817 --> 00:01:57.997
But it says Fabiano wants
to score at least 6.5,

00:01:57.997 --> 00:02:00.458
so that's greater than or equal to 6.5.

00:02:00.458 --> 00:02:02.663
He's ok if he scores 6.5.

00:02:02.663 --> 00:02:05.264
And there you have it,
we have our inequality

00:02:05.264 --> 00:02:08.898
in terms of the number of games he needs

00:02:08.898 --> 00:02:12.984
to win and draw and
this inequality needs to

00:02:12.984 --> 00:02:16.757
be true in order for him to score at least

00:02:16.757 --> 00:02:19.990
six and a half points in
this major chess tournament.


Constraining solutions to two variable linear inequalities

WEBVTT
Kind: captions
Language: en

00:00:00.608 --> 00:00:02.326
- [Voiceover] "Which x-values
make the ordered pair

00:00:02.326 --> 00:00:04.520
"X, comma negative
seven, a solution of the

00:00:04.520 --> 00:00:06.262
"following inequality?"

00:00:06.262 --> 00:00:07.864
The inequality is two X minus seven Y is

00:00:07.864 --> 00:00:09.802
less than 25.

00:00:09.802 --> 00:00:11.660
And so they give us some
choices, and I encourage you

00:00:11.660 --> 00:00:13.320
to pause the video and see
if you can figure it out

00:00:13.320 --> 00:00:14.748
on your own.

00:00:14.748 --> 00:00:16.025
All right, now let's
work through it together.

00:00:16.025 --> 00:00:18.440
They're constraining that
Y is going to be equal to

00:00:18.440 --> 00:00:19.763
negative seven.

00:00:19.763 --> 00:00:21.203
And so if we make that
constraint, we can replace

00:00:21.203 --> 00:00:23.199
this Y with a negative seven.

00:00:23.199 --> 00:00:27.866
So we can rewrite the
inequality as two X minus seven

00:00:27.866 --> 00:00:31.604
times negative seven, since
we're constraining Y to be

00:00:31.604 --> 00:00:35.122
negative seven, is less than 25.

00:00:35.122 --> 00:00:38.802
And so this is going to be
two X minus negative 49,

00:00:38.802 --> 00:00:43.616
or two X plus 49, is less than 25.

00:00:43.616 --> 00:00:46.464
Now if I just wanna
isolate the X on one side,

00:00:46.464 --> 00:00:49.157
which we see for these
inequalities up here,

00:00:49.157 --> 00:00:52.756
so we could subtract 49 from both sides.

00:00:52.756 --> 00:00:56.018
So subtracting 49 from both sides, we get

00:00:56.018 --> 00:01:00.963
two X is less than, let's see, 49 minus 25

00:01:00.963 --> 00:01:03.506
would be positive 24, so this would be

00:01:03.506 --> 00:01:05.502
negative 24.

00:01:05.502 --> 00:01:08.498
Now to isolate the X, we
just divide both sides by two

00:01:08.498 --> 00:01:09.879
and we're not gonna change
the inequality, since we

00:01:09.879 --> 00:01:12.224
are multiplying or dividing
by a positive value.

00:01:12.224 --> 00:01:13.501
Positive two.

00:01:13.501 --> 00:01:17.274
So this is going to be X
is less than negative 12.

00:01:17.274 --> 00:01:19.956
And lucky for us, this is a choice.

00:01:19.956 --> 00:01:23.496
So as long as, if Y is
equal to negative seven,

00:01:23.496 --> 00:01:26.840
as long as X is less
than negative 12, we will

00:01:26.840 --> 00:01:29.603
satisfy this inequality.

00:01:29.603 --> 00:01:31.077
Let's do another one of these.

00:01:31.077 --> 00:01:34.316
And this one is a little bit more visual.

00:01:34.316 --> 00:01:37.253
So "Which Y-values make the ordered pair,"

00:01:37.253 --> 00:01:40.898
so in the last one we
constrained what Y was,

00:01:40.898 --> 00:01:43.475
and we figured out what X values would

00:01:43.475 --> 00:01:45.252
satisfy this inequality.

00:01:45.252 --> 00:01:46.761
Now we're going the other way around.

00:01:46.761 --> 00:01:49.036
We're constraining X and we're saying,

00:01:49.036 --> 00:01:51.474
what Y values would make
the ordered pair true?

00:01:51.474 --> 00:01:53.157
Or make it a solution?

00:01:53.157 --> 00:01:55.560
"Which Y values make the
ordered pair 5 comma Y

00:01:55.560 --> 00:01:57.917
"a solution of the
inequality represented by the

00:01:57.917 --> 00:01:58.834
graph below?"

00:01:58.834 --> 00:02:00.599
So they didn't give it to us
algebraically, they gave it

00:02:00.599 --> 00:02:01.992
to us visually.

00:02:01.992 --> 00:02:03.455
And so to be a solution,
that means we have to be

00:02:03.455 --> 00:02:05.009
in this blue area.

00:02:06.929 --> 00:02:10.176
So this pair, so negative five,

00:02:10.176 --> 00:02:12.119
comma six.

00:02:12.119 --> 00:02:15.579
That would be a solution to
the inequality being depicted.

00:02:15.579 --> 00:02:18.144
Something that sits exactly
on the line, this would

00:02:18.144 --> 00:02:21.720
not be a solution, because
notice the line, as you can see

00:02:21.720 --> 00:02:24.239
this lower boundary
line, is a dashed line.

00:02:24.239 --> 00:02:26.294
If it was filled in, than
anything on the line would be

00:02:26.294 --> 00:02:26.921
a solution.

00:02:26.921 --> 00:02:28.720
But since it's dashed, things on the line

00:02:28.720 --> 00:02:29.637
aren't solutions.

00:02:29.637 --> 00:02:32.655
It's only things that are
above the line are going to

00:02:32.655 --> 00:02:34.420
be solutions.

00:02:34.420 --> 00:02:37.438
So let's see what they're
asking us to think about.

00:02:37.438 --> 00:02:40.236
So they're saying, they're
constraining X to be

00:02:40.236 --> 00:02:41.641
equal to five.

00:02:41.641 --> 00:02:44.694
So X equals five is everything,

00:02:44.694 --> 00:02:46.180
let me see if I can draw this.

00:02:46.180 --> 00:02:51.056
X equals five is everything on

00:02:51.056 --> 00:02:53.482
that line right over there.

00:02:53.482 --> 00:02:55.583
Now, if we assume that X equals five,

00:02:55.583 --> 00:02:57.650
we're gonna be someplace on this line,

00:02:57.650 --> 00:02:59.612
how do we have to constrain Y to make sure

00:02:59.612 --> 00:03:01.945
that we are in the solution?

00:03:01.945 --> 00:03:05.010
Well, we have to constrain Y so

00:03:05.010 --> 00:03:07.947
that we are above the line, for

00:03:07.947 --> 00:03:12.347
X equaling five, just to be clear.

00:03:12.347 --> 00:03:15.201
So we have to be above,

00:03:19.471 --> 00:03:22.068
our possible points are gonna be the ones

00:03:22.068 --> 00:03:24.598
once again, we're
constraining X equals five,

00:03:24.598 --> 00:03:28.349
the possible points are the ones,

00:03:28.349 --> 00:03:31.428
that I'm showing in magenta.

00:03:31.428 --> 00:03:34.551
And actually I could
keep going if I want to.

00:03:34.551 --> 00:03:37.871
So Y is going to have to be greater than,

00:03:37.871 --> 00:03:39.636
it can't be greater
than or equal to seven,

00:03:39.636 --> 00:03:41.331
it has to be greater than seven.

00:03:41.331 --> 00:03:43.212
If it was greater than or equal to seven,

00:03:43.212 --> 00:03:45.092
we'd be including the point on the line,

00:03:45.092 --> 00:03:46.776
but I already talked about
this being a dashed line

00:03:46.776 --> 00:03:48.749
so we don't want to include
the points on the line.

00:03:48.749 --> 00:03:51.746
We only want to include
the points above the line.

00:03:54.046 --> 00:03:57.434
So Y is going to be greater than seven

00:03:57.434 --> 00:04:01.775
which is this choice right over there.

00:04:01.775 --> 00:04:05.072
If X is equal to five, as long as Y is

00:04:05.072 --> 00:04:08.114
greater than seven, we are going to be

00:04:08.114 --> 00:04:10.413
in the solution set.


Visually completing solutions of systems of two variables

WEBVTT
Kind: captions
Language: en

00:00:00.168 --> 00:00:02.861
- [Voiceover] Which x-values
make the ordered pair,

00:00:02.861 --> 00:00:06.704
x comma negative two, a
solution of the system of

00:00:06.704 --> 00:00:09.815
inequalities represented
by the graph below?

00:00:09.815 --> 00:00:11.719
So let's look at this.

00:00:11.719 --> 00:00:14.876
So we're constraining
ourselves to all of the points

00:00:14.876 --> 00:00:16.792
of the form x comma negative two

00:00:16.792 --> 00:00:18.545
which is another way of
saying we're going to

00:00:18.545 --> 00:00:22.550
constrain ourselves to
y equaling negative two.

00:00:22.550 --> 00:00:25.290
If we constrain ourselves
to y equaling negative two,

00:00:25.290 --> 00:00:28.958
what has to be true of x
in order for this point

00:00:28.958 --> 00:00:33.705
to be a solution to this
system of inequalities?

00:00:33.705 --> 00:00:35.261
And so I encourage you to pause the video;

00:00:35.831 --> 00:00:40.140
look at this graph here, and
then pick one of the choices.

00:00:40.140 --> 00:00:42.230
All right, now let's
work through it together.

00:00:42.230 --> 00:00:44.401
So let's just be very clear
of what's going on here.

00:00:44.401 --> 00:00:45.887
Let me pick some points.

00:00:45.887 --> 00:00:50.414
So this point right over
here, this is a solution to

00:00:50.414 --> 00:00:52.945
neither of the inequalities in our system.

00:00:52.945 --> 00:00:55.453
You can think of this as the green system

00:00:55.453 --> 00:00:57.124
and this as the blue system.

00:00:57.124 --> 00:00:58.646
In order to be a solution
set, you have to be in

00:00:58.646 --> 00:01:00.401
the shaded area for that system.

00:01:00.401 --> 00:01:02.964
So this point right over
here, it's in the solution set

00:01:02.964 --> 00:01:07.236
for neither of the
inequalities of the system.

00:01:07.236 --> 00:01:09.091
This point right over here,

00:01:10.321 --> 00:01:13.039
it would still not be in
the solution set for either

00:01:13.039 --> 00:01:16.131
because it's on a dashed green line.

00:01:16.131 --> 00:01:18.395
If this was a solid green
line, then it would be part of

00:01:18.395 --> 00:01:22.063
the green solution set, but
since it's a dashed green line,

00:01:22.063 --> 00:01:25.268
the line itself is not
part of the solution set.

00:01:25.268 --> 00:01:27.328
Now this point right over here,

00:01:29.208 --> 00:01:32.675
this point satisfies the green inequality,

00:01:32.675 --> 00:01:34.301
it's part of its solution set,

00:01:34.301 --> 00:01:36.636
but it does not satisfy
the blue inequality

00:01:36.636 --> 00:01:40.431
so it's not in the solution
set for the system.

00:01:40.431 --> 00:01:45.414
Now this point here, this
actually would satisfy both,

00:01:45.414 --> 00:01:47.095
and the reason why it satisfies both,

00:01:47.095 --> 00:01:51.367
it clearly is in the shaded
area for the green inequality

00:01:52.017 --> 00:01:54.781
but it sits on the line
for the blue inequality,

00:01:54.781 --> 00:01:57.428
but that's okay because we're
including the line itself

00:01:57.428 --> 00:01:58.499
in the solution set;

00:01:59.574 --> 00:02:01.404
it's a solid blue line.

00:02:01.404 --> 00:02:03.633
So this would be in the solution set for

00:02:03.633 --> 00:02:05.278
the system of inequalities;

00:02:05.278 --> 00:02:06.774
this would be in the solution set for

00:02:06.774 --> 00:02:08.701
the system of inequalities,
all of these points,

00:02:08.701 --> 00:02:12.195
because they're in the solution
set of the blue inequality

00:02:12.195 --> 00:02:14.029
that we're seeing visually,
and in the green one.

00:02:14.029 --> 00:02:16.885
We assume that the green
one just keeps going down.

00:02:16.885 --> 00:02:18.209
And this is actually
what we're seeing here

00:02:18.209 --> 00:02:20.217
is actually the overlap.

00:02:20.217 --> 00:02:22.771
Now that we have a better
understanding of things,

00:02:22.771 --> 00:02:23.920
let's actually tackle the questions.

00:02:23.920 --> 00:02:27.206
We're constraining ourselves
to y equals negative two.

00:02:27.206 --> 00:02:29.597
So actually, let me draw
a line here that shows all

00:02:29.597 --> 00:02:32.708
of the points where y is
equal to negative two.

00:02:32.708 --> 00:02:35.135
So that shows, at least on our graph

00:02:35.135 --> 00:02:38.130
where y is equal to negative two.

00:02:38.130 --> 00:02:42.054
So given y equals negative
two, what has to be true of x

00:02:42.054 --> 00:02:47.007
in order to satisfy the
system of inequalities here?

00:02:47.557 --> 00:02:49.948
Well, we're going to have to
deal with all of the x-values

00:02:50.588 --> 00:02:52.658
including and to the right of this point.

00:02:54.004 --> 00:02:57.979
I can say including because
the blue inequality,

00:02:59.149 --> 00:03:00.888
you can also be equal to the line;

00:03:00.888 --> 00:03:02.896
you can actually be on the line.

00:03:02.896 --> 00:03:05.996
So being on the line is part
of the solution set for both

00:03:05.996 --> 00:03:09.328
or anything to the right.

00:03:10.490 --> 00:03:12.166
So all of this

00:03:14.378 --> 00:03:17.618
is part of the solution set.

00:03:17.618 --> 00:03:19.880
And so, if we constrain
y equals negative two,

00:03:21.694 --> 00:03:24.247
we see that x has to be greater than

00:03:24.247 --> 00:03:27.755
or equal to negative three.

00:03:27.759 --> 00:03:28.940
And we see that

00:03:30.456 --> 00:03:33.433
that is this choice,

00:03:33.433 --> 00:03:34.880
this choice right over there,

00:03:34.880 --> 00:03:38.481
x is greater than or
equal to negative three.

00:03:38.481 --> 00:03:41.136
Now let's do another one and
instead of constraining y,

00:03:41.136 --> 00:03:44.195
we're now going to constrain x.

00:03:44.195 --> 00:03:48.594
Which y-values make the
ordered pair, four comma y,

00:03:48.594 --> 00:03:51.403
a solution of the system
of inequalities represented

00:03:51.403 --> 00:03:53.110
by the graph below?

00:03:53.110 --> 00:03:55.336
And once again, I encourage
you to pause the video

00:03:55.336 --> 00:03:58.482
and see if you can work
through it on your own.

00:03:58.482 --> 00:03:59.727
All right now, let's
work through it together.

00:04:00.387 --> 00:04:03.420
In this scenario, we are constraining x.

00:04:03.420 --> 00:04:06.293
We're saying that x has
to be equal to four.

00:04:06.293 --> 00:04:07.529
So x equaling four,

00:04:10.452 --> 00:04:13.601
that's all of the points on this line

00:04:13.601 --> 00:04:15.749
right over here.

00:04:15.749 --> 00:04:18.002
So, we are constraining
ourselves to the points

00:04:18.002 --> 00:04:20.300
that sit on this line,
but we want to be part

00:04:20.300 --> 00:04:22.088
of the solution set.

00:04:22.088 --> 00:04:25.420
So we want to be on this line

00:04:25.420 --> 00:04:27.312
that constrains us to x equals four,

00:04:27.312 --> 00:04:30.087
but we want to be in the
overlap of the solution sets

00:04:30.087 --> 00:04:33.041
of the two inequalities in
order to satisfy the system.

00:04:33.041 --> 00:04:34.433
So, let's see.

00:04:34.433 --> 00:04:38.598
We want to be in this
area right over here,

00:04:38.598 --> 00:04:41.279
that's the overlap of the solutions sets

00:04:41.279 --> 00:04:43.525
of the two inequalities.

00:04:43.525 --> 00:04:46.930
And so, if we constrain
ourselves to x equaling four,

00:04:48.600 --> 00:04:51.787
y has to be greater than
because we're not including

00:04:51.787 --> 00:04:54.259
the green line itself; it's dashed.

00:04:54.259 --> 00:04:58.439
So y has to be greater than negative one,

00:04:58.439 --> 00:05:01.770
or we can say negative
one has to be less than y,

00:05:01.770 --> 00:05:04.242
and then y can go all the way

00:05:05.450 --> 00:05:08.528
up to and including three.

00:05:08.528 --> 00:05:11.036
Up to and including three
because this blue line

00:05:11.036 --> 00:05:12.278
is actually filled in.

00:05:12.278 --> 00:05:14.948
So anything that's on the
blue line is still going

00:05:14.948 --> 00:05:18.233
to be on the solution set
of the blue inequality.

00:05:18.233 --> 00:05:20.497
And this point that I'm
showing right here is clearly

00:05:20.497 --> 00:05:23.736
sitting in the overlap
for both inequalities,

00:05:23.736 --> 00:05:28.554
and so y has to be less
than or equal to three.

00:05:28.554 --> 00:05:31.290
So if x is equal to four, y has to be

00:05:32.920 --> 00:05:35.560
greater than negative one and less than

00:05:35.560 --> 00:05:36.961
or equal to negative three...

00:05:36.961 --> 00:05:39.260
And less than or equal
to three, I should say.

00:05:39.260 --> 00:05:40.724
So let's see which of these choices:

00:05:40.724 --> 00:05:42.914
negative one is less than y is
less than or equal to three.

00:05:42.914 --> 00:05:45.303
That's, once again, our first choice.


System of equations word problem: walk & ride

WEBVTT
Kind: captions
Language: en

00:00:00.239 --> 00:00:03.002
- [Voiceover] Yochanan walked
from home to the bus stop

00:00:03.002 --> 00:00:06.517
at an average speed of
five kilometers per hour.

00:00:06.517 --> 00:00:08.406
He immediately got on his school bus

00:00:08.406 --> 00:00:10.133
and traveled at an average speed

00:00:10.133 --> 00:00:13.668
of 60 kilometers per hour
until he got to school.

00:00:14.383 --> 00:00:16.374
The total distance from his home to school

00:00:16.374 --> 00:00:18.526
is 35 kilometers, and the entire trip took

00:00:18.526 --> 00:00:20.240
one and a half hours.

00:00:20.559 --> 00:00:22.286
How many kilometers did Yochanan cover

00:00:22.286 --> 00:00:24.094
by walking, and how many kilometers

00:00:24.094 --> 00:00:27.137
did he cover by travelling
on the bus? It's fascinating.

00:00:27.137 --> 00:00:28.620
So I encourage you to pause the video

00:00:28.620 --> 00:00:31.037
and try to think about on your own.

00:00:31.037 --> 00:00:32.785
Alright, so let's just
define some variables here.

00:00:32.785 --> 00:00:34.837
How many kilometers did
he cover by walking?

00:00:34.837 --> 00:00:38.230
Let's call this W and how many kilometers

00:00:38.230 --> 00:00:41.388
did he cover by travelling on the bus?

00:00:41.388 --> 00:00:44.284
Let's call that, Let's call that B

00:00:45.154 --> 00:00:46.094
and so what do we know?

00:00:46.094 --> 00:00:47.812
We know that the kilometers

00:00:47.812 --> 00:00:49.243
and I can actually even
draw it just to make sure

00:00:49.243 --> 00:00:51.104
that we're visualizing this thing, right?

00:00:51.222 --> 00:00:55.578
So this right over here is
his home, that's his home

00:00:55.578 --> 00:00:59.627
and then he is going to
travel and see they tell us,

00:00:59.865 --> 00:01:02.935
they tell us that it is
35 kilometers to school

00:01:02.935 --> 00:01:05.618
so this is his school right over here.

00:01:05.618 --> 00:01:08.564
I'll draw a bigger
building, that's his school,

00:01:08.564 --> 00:01:13.420
and we know that this
distance is 35, 35 kilometers

00:01:13.420 --> 00:01:15.719
and we also know that it
took one and a half hours.

00:01:16.325 --> 00:01:17.910
One and a half hours, now he traveled

00:01:17.910 --> 00:01:19.657
at different rates for
different distances.

00:01:19.657 --> 00:01:23.678
So he traveled some
distance to the school bus

00:01:24.505 --> 00:01:26.862
so this is, or to the bus stop,

00:01:26.862 --> 00:01:30.437
so that's the bus stop, right over there,

00:01:30.437 --> 00:01:32.611
and we're seeing this
distance to the bus stop,

00:01:32.611 --> 00:01:34.696
that's how much he covered by walking.

00:01:34.696 --> 00:01:37.053
So this right over here, this
distance, right over here,

00:01:37.053 --> 00:01:40.141
that is W and the rest of the distance,

00:01:40.141 --> 00:01:43.839
he covered by the bus, so
the rest of this distance,

00:01:43.839 --> 00:01:45.876
all of this distance right over there,

00:01:45.876 --> 00:01:48.468
that is going to be B.

00:01:48.468 --> 00:01:49.452
So what do we know?

00:01:49.452 --> 00:01:51.464
We know the distance covered by walking

00:01:51.464 --> 00:01:53.008
plus the distance covered by bus

00:01:53.008 --> 00:01:56.554
is going to be 35 kilometers,
35 kilometers here,

00:01:56.554 --> 00:02:00.557
this is the entire, that
is the entire distance

00:02:00.557 --> 00:02:03.997
from home to school, so
we know that W plus B,

00:02:05.560 --> 00:02:10.560
W plus B, plus B is
equal to 35 kilometers,

00:02:13.746 --> 00:02:17.403
is equal to 35 kilometers
and just with one equation,

00:02:17.403 --> 00:02:19.468
we're not going to be able to
figure out what W and B are

00:02:19.468 --> 00:02:20.527
but we have another constraint.

00:02:20.527 --> 00:02:22.631
We know the total amount of time.

00:02:22.631 --> 00:02:24.805
So the total amount of time is going to be

00:02:24.805 --> 00:02:27.934
one and half hours, so we'll
just write that over here.

00:02:27.934 --> 00:02:32.934
This is going to be 1.5, so
what's the time traveled by,

00:02:33.268 --> 00:02:35.256
what's the time he walks?

00:02:35.818 --> 00:02:38.827
Let me write this over here, time

00:02:38.827 --> 00:02:43.477
time walking, we'll that's going to be

00:02:43.477 --> 00:02:46.334
the distance walking
divided by the rate walking.

00:02:46.334 --> 00:02:50.003
So the distance walking is W kilometers

00:02:50.003 --> 00:02:55.003
W kilometers divided by his rate,

00:02:55.285 --> 00:02:56.322
the distance divided by your rate

00:02:56.322 --> 00:02:58.556
is gonna give you your time, so let's see,

00:02:58.556 --> 00:03:00.913
his rate is five kilometers per hour,

00:03:00.913 --> 00:03:05.078
five kilometers per
hour and so you're gonna

00:03:05.078 --> 00:03:06.419
have kilometers cancel kilometers

00:03:06.419 --> 00:03:09.507
and if you divide by or
if you have one over hours

00:03:09.507 --> 00:03:12.453
in the denominator, that's
going to be the same thing,

00:03:12.453 --> 00:03:17.390
this is gonna be W over five
hours, so the units work out.

00:03:17.390 --> 00:03:20.397
So his time walking is W over five,

00:03:20.397 --> 00:03:23.424
W over five and by that same logic,

00:03:23.424 --> 00:03:25.802
his time on the bus is
going to be the distance

00:03:25.802 --> 00:03:30.353
on the bus divided by,
divided by the average speed

00:03:30.353 --> 00:03:33.278
of the school bus, so
this is going to be 60.

00:03:33.278 --> 00:03:35.188
This is all going to be in hours

00:03:35.188 --> 00:03:37.829
and now we can solve
this system of equations.

00:03:37.829 --> 00:03:40.999
We have two linear
equations with two unknowns.

00:03:40.999 --> 00:03:42.604
We should be able to find W and B

00:03:42.604 --> 00:03:45.631
that satisfy both of these.

00:03:45.631 --> 00:03:48.232
Now what's an easy thing to do?

00:03:48.232 --> 00:03:50.343
Let's see, if I can multiply
this second equation

00:03:50.343 --> 00:03:53.972
by negative five, and I'm gonna,

00:03:53.972 --> 00:03:55.355
this is going to be a negative W here

00:03:55.355 --> 00:03:57.671
so it'll cancel out with this W up there.

00:03:57.671 --> 00:04:00.800
So let's do that, let's
multiply the second equation

00:04:00.800 --> 00:04:03.461
by, I'm just gonna
switch to one color here,

00:04:03.461 --> 00:04:08.461
so this top equation is going
to be W plus B is equal to 35.

00:04:09.325 --> 00:04:11.580
This bottom equation,
if I multiply both sides

00:04:11.580 --> 00:04:15.388
by negative five, so both
sides by negative five,

00:04:15.388 --> 00:04:19.526
I'm going to multiply both
sides by negative five

00:04:19.526 --> 00:04:22.411
I'm going to get negative
five times W over five

00:04:22.411 --> 00:04:27.411
is negative W, negative
five times B over 60.

00:04:27.835 --> 00:04:29.176
Let's see, it's gonna be,
it's going to be negative five

00:04:29.176 --> 00:04:34.032
over 60, that's negative
1/12, so this is negative B

00:04:34.032 --> 00:04:38.644
over 12 and then it is
going to be equal to

00:04:38.644 --> 00:04:43.644
1.5 times negative five is
negative 7.5, negative 7.5.

00:04:45.186 --> 00:04:46.567
Now we can just add the left and

00:04:46.567 --> 00:04:49.147
right hand sides of these two equations.

00:04:49.147 --> 00:04:51.403
Now let me, I can do
this a little bit neater,

00:04:51.403 --> 00:04:54.674
let me actually delete, let me make these

00:04:54.674 --> 00:04:58.493
line up a little bit better
so that we, delete that,

00:04:58.493 --> 00:05:02.394
make this, so this first
equation was, whoops,

00:05:02.394 --> 00:05:07.394
this first equation was
W plus B is equal to 35

00:05:10.095 --> 00:05:11.674
now they line up better and now we can add

00:05:11.674 --> 00:05:13.830
the left hand sides of these equations

00:05:13.830 --> 00:05:16.016
and the right hand sides,
so the left hand sides,

00:05:16.016 --> 00:05:18.162
the W's cancel out. That's what we wanted.

00:05:18.162 --> 00:05:22.530
Now B minus B over 12 or
B minus the twelfth of B,

00:05:22.530 --> 00:05:24.582
well that's going to be 11/12B,

00:05:24.582 --> 00:05:28.930
11/12B is equal to, let's see,

00:05:28.930 --> 00:05:33.930
35 minus 7.5, see 35 minus seven

00:05:36.510 --> 00:05:40.604
would be 28 and then another
half this would be 27.5.

00:05:41.678 --> 00:05:45.497
27.5 and since I'm dealing
with a fraction here,

00:05:45.497 --> 00:05:47.123
let me write this as a fraction

00:05:47.123 --> 00:05:51.753
so this is the same thing as 55 over two.

00:05:51.970 --> 00:05:53.799
Let's me just write it this way.

00:05:53.799 --> 00:05:58.058
This is the same thing as 55 over two.

00:05:58.058 --> 00:05:59.703
Now to solve for B, I
just have to multiply

00:05:59.703 --> 00:06:01.447
both sides times this reciprocal.

00:06:01.447 --> 00:06:04.190
I'll switch colors just
to ease the monotony.

00:06:04.190 --> 00:06:09.190
So multiply both sides by
12 over 11, 12 over 11,

00:06:10.712 --> 00:06:13.414
what we get, these cancel out, what we get

00:06:13.414 --> 00:06:16.543
is that B, so I'll do this in this color,

00:06:16.543 --> 00:06:20.097
B is equal to, let's see, I
have a 12 in the numerator

00:06:20.097 --> 00:06:21.459
two in the denominator so I can

00:06:21.459 --> 00:06:25.542
make that a six and a one,
then I have a 55 and 11,

00:06:25.542 --> 00:06:29.151
I can divide both by 11, so
it's gonna be five and a one.

00:06:29.151 --> 00:06:31.650
So it's five times six, B is equal to 30

00:06:31.650 --> 00:06:35.868
and B was in kilometers so
if he travels 30 kilometers,

00:06:35.868 --> 00:06:39.606
30 kilometers by bus, B
is equal to 30 kilometers,

00:06:39.606 --> 00:06:42.288
and the amount that he walks,
well, we can figure that out.

00:06:42.288 --> 00:06:43.853
If this is 30, well the amount

00:06:43.853 --> 00:06:46.472
that he walks is going to be five.

00:06:46.892 --> 00:06:50.529
This is going to be five kilometers, W,

00:06:50.529 --> 00:06:53.475
let me write this, W is
equal to five kilometers.

00:06:53.475 --> 00:06:55.141
He walks five kilometers and then he goes

00:06:55.141 --> 00:06:58.916
by the bus 30 kilometers.


System of equations word problem: infinite solutions

WEBVTT
Kind: captions
Language: en

00:00:00.634 --> 00:00:02.514
- [Voiceover] Farmer Jan
is a vegetable farmer

00:00:02.514 --> 00:00:04.998
who divides his field
between broccoli crops

00:00:04.998 --> 00:00:06.531
and spinach crops.

00:00:06.531 --> 00:00:10.083
Last year, he grew six
tons of broccoli per acre,

00:00:10.083 --> 00:00:12.869
so six tons of broccoli per acre,

00:00:12.869 --> 00:00:15.748
and nine tons of spinach,

00:00:15.748 --> 00:00:17.931
nine tons of spinach per acre,

00:00:17.931 --> 00:00:21.414
for a total of 93 tons of vegetables.

00:00:21.414 --> 00:00:24.966
This year, he grew two
tons of broccoli per acre,

00:00:24.966 --> 00:00:29.169
so this year, he grew two
tons of broccoli per acre,

00:00:29.169 --> 00:00:32.559
and three tons of spinach per acre,

00:00:32.559 --> 00:00:35.229
and three tons of spinach per acre,

00:00:35.229 --> 00:00:38.458
for a total of 31 tons of vegetables.

00:00:38.458 --> 00:00:41.800
How many acres of broccoli
crops and how many acres

00:00:41.800 --> 00:00:45.028
of spinach crops does Farmer Jan have?

00:00:45.028 --> 00:00:46.467
So let's think about this.

00:00:46.467 --> 00:00:51.042
So let's say that the number
of acres of broccoli crops,

00:00:51.042 --> 00:00:54.385
let's call that B,

00:00:54.385 --> 00:00:57.589
and let's say the acres of spinach crops,

00:00:57.589 --> 00:01:01.537
acres of spinach crops, let's call that S.

00:01:01.537 --> 00:01:04.811
So how much broccoli is he going to grow

00:01:04.811 --> 00:01:07.388
in, I guess you could say, last year?

00:01:07.388 --> 00:01:09.385
Last year, how much broccoli did he grow?

00:01:09.385 --> 00:01:11.576
And let me just write this
down. This is last year.

00:01:11.576 --> 00:01:15.019
Last year.

00:01:15.019 --> 00:01:19.241
And they tell us, last
year he grew six tons

00:01:19.241 --> 00:01:21.453
of broccoli per acre.

00:01:21.453 --> 00:01:23.914
So if he grew six tons
of broccoli per acre,

00:01:23.914 --> 00:01:25.935
and he has B acres,

00:01:25.935 --> 00:01:29.860
well, that means he grew six tons per acre

00:01:29.860 --> 00:01:31.903
times B acres.

00:01:31.903 --> 00:01:36.499
So he grew 6B tons of broccoli last year.

00:01:36.499 --> 00:01:40.262
And by the same logic,
he grew how much spinach?

00:01:40.262 --> 00:01:45.022
Well, nine tons of spinach
per acre times S acres.

00:01:45.022 --> 00:01:49.311
So 9S tons of spinach,

00:01:49.311 --> 00:01:51.957
and then the total is 93 tons,

00:01:51.957 --> 00:01:55.534
so for a total of 93 tons of vegetables.

00:01:55.534 --> 00:01:59.132
So this is going to be equal to 93.

00:01:59.132 --> 00:02:01.315
So now let's think about this year.

00:02:01.315 --> 00:02:03.892
And in general, when you
tackle these, just think about,

00:02:03.892 --> 00:02:05.889
well, set the variables in general

00:02:05.889 --> 00:02:07.405
to what they're asking for, and then,

00:02:07.405 --> 00:02:10.181
how can you use the information
that they're giving us?

00:02:10.181 --> 00:02:13.477
So this year, how much
broccoli would he have grown?

00:02:13.477 --> 00:02:16.542
Well, he grew two tons
of broccoli per acre,

00:02:16.542 --> 00:02:19.701
so he grew two tons per acre.

00:02:19.701 --> 00:02:22.673
He has the same number of
acres. We can assume that.

00:02:22.673 --> 00:02:24.785
So two tons per acre times B acres

00:02:24.785 --> 00:02:28.106
is gonna be 2B tons of broccoli.

00:02:28.106 --> 00:02:30.242
And by that same logic, it's going to be,

00:02:30.242 --> 00:02:32.610
he grew three tons of spinach per acre.

00:02:32.610 --> 00:02:34.283
Well, he has S acres.

00:02:34.283 --> 00:02:36.651
Each of those acres,
he's growing three tons

00:02:36.651 --> 00:02:41.017
of spinach per acre, so it's
going to be 3S tons of spinach.

00:02:41.017 --> 00:02:42.827
And they tell us what that total is.

00:02:42.827 --> 00:02:47.395
That total is, for a total
of 31 tons of vegetables.

00:02:47.395 --> 00:02:50.112
So this is going to be equal to 31.

00:02:50.112 --> 00:02:52.456
And so now we have a
system of two equations

00:02:52.456 --> 00:02:56.962
with two unknowns, that we
can use to solve for B and S.

00:02:56.962 --> 00:03:00.026
So let's see. What do we
wanna solve for first?

00:03:00.026 --> 00:03:03.787
Well, what we could do, let
me rewrite the top equation.

00:03:03.787 --> 00:03:07.028
So we have 6B plus,

00:03:07.868 --> 00:03:12.261
plus 9S is equal to 93,

00:03:13.211 --> 00:03:15.321
is equal to 93.

00:03:15.321 --> 00:03:16.299
And the second equation,

00:03:16.299 --> 00:03:20.084
we'll let's try to eliminate the Bs.

00:03:20.084 --> 00:03:24.326
So let's multiply the second
equation by negative three.

00:03:24.326 --> 00:03:27.791
I'm gonna multiply the left
hand side by negative three,

00:03:27.791 --> 00:03:30.253
and I'm gonna multiply the right hand side

00:03:30.253 --> 00:03:31.623
by negative three.

00:03:31.623 --> 00:03:34.014
What am I going to get?

00:03:34.014 --> 00:03:37.335
Negative three times 2B is negative 6B.

00:03:37.335 --> 00:03:38.914
That was the whole point by multiplying it

00:03:38.914 --> 00:03:40.585
by negative three.

00:03:40.585 --> 00:03:44.685
Negative three times 3S is negative 9S,

00:03:44.685 --> 00:03:46.242
negative 9S.

00:03:46.242 --> 00:03:51.140
And then, negative three times 31 is going

00:03:51.140 --> 00:03:54.159
to be negative 93.

00:03:54.159 --> 00:03:57.688
So what do we get if we
now add the two sides

00:03:57.688 --> 00:03:59.477
of these equations?

00:03:59.477 --> 00:04:02.612
So on the left hand side,
6B minus 6B, that's zero.

00:04:02.612 --> 00:04:06.117
9S minus 9S, that's zero.
We're gonna get just zero.

00:04:06.117 --> 00:04:10.296
On the right hand side we get 93 minus 93.

00:04:10.296 --> 00:04:13.454
Well, that's still just going to be zero.

00:04:13.454 --> 00:04:16.356
So we have this situation
where we get zero equals zero,

00:04:16.356 --> 00:04:19.421
which is going to be true
no matter what X and Y are,

00:04:19.421 --> 00:04:22.092
and so this is a system
with an infinite number

00:04:22.092 --> 00:04:23.253
of solutions.

00:04:23.253 --> 00:04:26.482
So this has infinite,

00:04:27.592 --> 00:04:31.655
infinite number of solutions.

00:04:31.655 --> 00:04:33.491
So one way we could think about it is,

00:04:33.491 --> 00:04:34.861
these two constraints,

00:04:34.861 --> 00:04:37.460
they're not giving us enough information.

00:04:37.460 --> 00:04:40.317
There's an infinite number
of B and S combinations

00:04:40.317 --> 00:04:42.987
that would satisfy these equations,

00:04:42.987 --> 00:04:45.565
so they're not giving
us enough information

00:04:45.565 --> 00:04:50.325
to say exactly what B and S are.

00:04:50.325 --> 00:04:53.617
So there is not enough,

00:04:53.617 --> 00:04:56.495
not enough info!

00:04:56.495 --> 00:04:58.492
Put an exclamation mark there.


System of equations word problem: no solution

WEBVTT
Kind: captions
Language: en

00:00:00.629 --> 00:00:04.065
- [Voiceover] A factory has
machines that produce toys,

00:00:04.065 --> 00:00:06.642
which are then packed by
the factory's workers.

00:00:06.642 --> 00:00:09.986
One day, each machine produced 14 toys

00:00:09.986 --> 00:00:12.424
and each worker packed two toys,

00:00:12.424 --> 00:00:16.836
so that a total of 40
toys remained unpacked.

00:00:16.836 --> 00:00:20.736
Additionally, the number of
workers that day was eight less

00:00:20.736 --> 00:00:23.499
than seven times the number of machines.

00:00:23.499 --> 00:00:26.402
How many machines and workers were there?

00:00:26.402 --> 00:00:27.586
I encourage you to pause the video.

00:00:27.586 --> 00:00:30.117
This is a good little problem over here.

00:00:30.117 --> 00:00:32.648
All right, so let's define some variables.

00:00:32.648 --> 00:00:34.965
So let's say M is equal

00:00:34.965 --> 00:00:38.536
to the number of machines,

00:00:39.586 --> 00:00:43.451
and let's say that W is equal

00:00:43.451 --> 00:00:47.723
to the number of workers.

00:00:47.723 --> 00:00:49.441
Those seem like reasonable variables.

00:00:49.441 --> 00:00:51.614
So what does the first sentence tell us?

00:00:51.614 --> 00:00:56.593
It tells us one day each
machine produced 14 toys.

00:00:56.593 --> 00:00:59.240
So if each machine produced 14 toys,

00:00:59.240 --> 00:01:00.842
what is the total number of toys

00:01:00.842 --> 00:01:02.630
that are going to be produced?

00:01:02.630 --> 00:01:03.907
Well, the total number
of toys that are going

00:01:03.907 --> 00:01:07.808
to be produced is going to
be the number per machine

00:01:07.808 --> 00:01:10.292
times the number of machines.

00:01:10.292 --> 00:01:15.121
So this is 14M toys produced.

00:01:15.121 --> 00:01:19.946
So this is what is produced.
Produced, right over there.

00:01:19.946 --> 00:01:22.743
And then how many toys
are going to be packed?

00:01:22.743 --> 00:01:25.437
Well, if each worker packed two toys,

00:01:25.437 --> 00:01:28.780
they tell us there, each
worker packed two toys,

00:01:28.780 --> 00:01:30.684
so the total number that's
going to be packed is going

00:01:30.684 --> 00:01:34.551
to be two toys per worker
times the number of workers.

00:01:34.551 --> 00:01:37.598
So that right over there, that's
the number of toys packed.

00:01:38.898 --> 00:01:42.143
And then they tell us the
number of toys that remain,

00:01:42.143 --> 00:01:45.510
the total that remains unpacked.

00:01:45.510 --> 00:01:49.201
So the total that remains
unpacked, we know that that is 40.

00:01:49.201 --> 00:01:52.290
Let me do that in a neutral color.

00:01:52.290 --> 00:01:55.751
So 40, for 40,

00:01:55.751 --> 00:01:59.148
so that we could view the 40
as produced, but not packed.

00:01:59.148 --> 00:02:04.094
Produced, not packed.

00:02:04.094 --> 00:02:07.599
That's the number, the
total the remain unpacked.

00:02:07.599 --> 00:02:10.642
Well, how do we relate produced and packed

00:02:10.642 --> 00:02:12.452
to the produced that are not packed?

00:02:12.452 --> 00:02:14.147
Well, if we take the
total that were produced,

00:02:14.147 --> 00:02:16.609
we subtract out the
number that were packed,

00:02:16.609 --> 00:02:19.418
we're gonna be left with
the total that are unpacked.

00:02:19.418 --> 00:02:20.254
So just like that,

00:02:20.254 --> 00:02:25.246
we're able to set up a linear
relationship between M and W.

00:02:25.246 --> 00:02:27.939
Well, just one isn't enough
to solve for M and W,

00:02:27.939 --> 00:02:29.518
but we have another relationship.

00:02:29.518 --> 00:02:32.839
They say, additionally, the
number of workers that day,

00:02:32.839 --> 00:02:35.346
so the number of workers that day.

00:02:35.346 --> 00:02:38.110
I could say W. I'll write it over here.

00:02:38.110 --> 00:02:40.222
W, the number of workers that day,

00:02:40.222 --> 00:02:43.543
was eight less than seven
times the number of machines.

00:02:43.543 --> 00:02:47.281
Or you could say it was equal
to seven times the number

00:02:47.281 --> 00:02:49.788
of machines minus eight.

00:02:49.788 --> 00:02:52.250
That would be eight less
than seven times the number

00:02:52.250 --> 00:02:53.620
of machines.

00:02:53.620 --> 00:02:56.127
7M minus eight.

00:02:56.127 --> 00:02:59.286
And now we have two
equations with two unknowns.

00:02:59.286 --> 00:03:00.958
If things work out well, we might be able

00:03:00.958 --> 00:03:04.811
to actually solve for W and M.

00:03:04.811 --> 00:03:06.600
So there is a bunch of ways to do it.

00:03:06.600 --> 00:03:11.452
Since this equation already
has W explicitly solved for,

00:03:11.452 --> 00:03:13.287
we can do some substitution here.

00:03:13.287 --> 00:03:17.211
We can take this W and
substitute it in for this W.

00:03:17.211 --> 00:03:21.292
Or, actually I should say,
we could take 7M minus eight

00:03:21.292 --> 00:03:23.712
and substitute it in for this W,

00:03:23.712 --> 00:03:26.754
since the M and W, the
pair that we wanna find,

00:03:26.754 --> 00:03:28.635
need to satisfy both equations.

00:03:28.635 --> 00:03:30.770
And so we are going to get,

00:03:30.770 --> 00:03:35.046
we're going to get 14M minus,

00:03:35.046 --> 00:03:37.988
minus two,

00:03:37.988 --> 00:03:41.643
minus two times, let me do that in a,

00:03:41.643 --> 00:03:46.009
so minus two times, and instead of a W,

00:03:46.009 --> 00:03:48.280
I can write the 7M minus eight.

00:03:49.260 --> 00:03:53.085
So 7M minus eight,

00:03:54.025 --> 00:03:55.837
and we get that equals 40,

00:03:55.837 --> 00:03:58.392
is equal to 40, so we get,

00:03:58.392 --> 00:04:00.133
now it's just a little bit of algebra.

00:04:00.133 --> 00:04:02.339
14M and then, let's see,

00:04:02.339 --> 00:04:04.288
I'll do everything in a neutral color now.

00:04:04.288 --> 00:04:06.945
So negative two times 7M

00:04:06.945 --> 00:04:10.035
is negative 14M,

00:04:10.035 --> 00:04:13.344
and then negative two times negative eight

00:04:13.344 --> 00:04:16.134
is plus 16,

00:04:16.134 --> 00:04:18.987
and then that's going to be equal to 40.

00:04:18.987 --> 00:04:22.515
Now, 14M minus 14M, that's
just going to be zero,

00:04:22.515 --> 00:04:26.219
and we're left that 16 is equal to 40.

00:04:26.219 --> 00:04:28.251
Well, that's never going to be true.

00:04:28.251 --> 00:04:32.267
16 is never going to be equal to 40.

00:04:32.267 --> 00:04:35.380
Doesn't matter what M and W are.

00:04:35.380 --> 00:04:37.446
In fact, M and W have been
eliminated from this equation.

00:04:37.446 --> 00:04:39.466
This is impossible.

00:04:39.466 --> 00:04:42.484
This right over here is impossible,

00:04:42.484 --> 00:04:44.341
for 16 to be equal to 40,

00:04:44.341 --> 00:04:46.617
and because of that, there
are no solutions to this.

00:04:46.617 --> 00:04:50.240
There's no M and W pair

00:04:50.240 --> 00:04:52.747
that matches the constraints they gave us,

00:04:52.747 --> 00:04:56.277
so there is no solution.

00:04:56.277 --> 00:04:58.622
No solution.

00:04:58.622 --> 00:05:01.037
I'll put that in a little square there.


Number of solutions to a system of equations algebraically

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:04.920
how many solutions does the following
system of linear equations have and I

00:00:04.920 --> 00:00:08.370
have my system right over here there's
a couple of ways to think about it one

00:00:08.370 --> 00:00:11.700
way is to think about them graphically
and think about well are they the same

00:00:11.700 --> 00:00:14.310
line in which case they would have
an infinite number of solutions are

00:00:14.310 --> 00:00:17.730
they parallel in which case they never
intersect you'd have no solutions or do

00:00:17.730 --> 00:00:21.420
they intersect exactly one place in case
you would have one solution but instead

00:00:21.420 --> 00:00:24.810
we're going to do this algebraically
so let's try to actually just solve

00:00:24.810 --> 00:00:29.280
the system and see what we get so the
first equation I'm going to leave that

00:00:29.280 --> 00:00:36.990
unchanged 5x minus 9y is equal to
is equal to is 16 now this second

00:00:36.990 --> 00:00:40.590
equation right over here let's say I
want to cancel out the X terms so let me

00:00:40.590 --> 00:00:45.300
multiply the second equation by negative
one so I have a negative 5x that I can

00:00:45.300 --> 00:00:49.740
cancel out with the 5x so if I multiply
the second equation by negative one I'm

00:00:49.740 --> 00:00:59.580
going to have negative 5x plus 9y plus
9 Y is equal to negative 36 negative 36

00:00:59.580 --> 00:01:02.910
now what I'm going to do is I'm going
to add the left side of the equations

00:01:02.910 --> 00:01:08.070
and the right sides of the equations to
get a new equation so 5x minus 5x well

00:01:08.070 --> 00:01:13.380
that's going to be 0 negative 9y plus 9y
well that's going to be 0 again I don't

00:01:13.380 --> 00:01:17.040
even have to write it it's just going
to be 0 on the left-hand side and on

00:01:17.040 --> 00:01:22.320
the right-hand side I'm going to have
16 minus 36 so negative 20 so now I'm

00:01:22.320 --> 00:01:27.150
left with the somewhat bizarre-looking
equation that says that 0 is equal to

00:01:27.150 --> 00:01:31.200
negative 20 now one way you might say
well how does this make any sense and

00:01:31.200 --> 00:01:35.130
the way to think about it is well are
there any X Y values 4 which is 0 is

00:01:35.130 --> 00:01:40.560
going to be equal to negative 20 well no
0 is never going to be equal to negative

00:01:40.560 --> 00:01:45.540
20 and so it doesn't matter what X Y
values you can never find an X well X or

00:01:45.540 --> 00:01:50.010
XY pair that's going to make 0 equal to
negative 20 in fact the X's and Y's have

00:01:50.010 --> 00:01:54.240
disappeared from this equation there's
no way that this is going to be true so

00:01:54.240 --> 00:02:00.060
we have no we have no solutions now if
you were to plot these if you were to

00:02:00.060 --> 00:02:03.660
plot each of these lines you would see
that they are parallel lines and that's

00:02:03.660 --> 00:02:08.430
why they have the same slope different
y-intercepts and that's why we have no

00:02:08.430 --> 00:02:12.540
solutions they don't intersect let's do
another one of these this is this is fun

00:02:12.680 --> 00:02:17.450
all right how many solutions does the
following system of linear equations

00:02:17.450 --> 00:02:22.430
have so let's do the same thing I'm
going to keep the first equation the

00:02:22.430 --> 00:02:29.570
same negative 6x plus 4y is equal to 2
and the second equation let me just see

00:02:29.570 --> 00:02:34.310
if I can cancel out the X term so if I
have a negative 6x if I multiply this by

00:02:34.310 --> 00:02:38.360
2 I'm going to have a positive 6x so I
can let's see I'm going to multiply this

00:02:38.360 --> 00:02:44.060
whole equation both sides of it by 2 so
I'm going to have 6x 3 x times 2 is 6x

00:02:44.060 --> 00:02:51.680
negative 2y times 2 is negative 4y and
that is going to be equal to negative

00:02:51.680 --> 00:02:56.330
2 now let's do the same thing let's
add the left sides and let's add the

00:02:56.330 --> 00:03:02.720
right sides so negative 6x plus 6x well
that's going to be 0 4y minus 4y that's

00:03:02.720 --> 00:03:06.320
0 we just have a 0 on the left-hand
side now on the right-hand side we

00:03:06.320 --> 00:03:10.370
have 2 plus negative 2 well that's 0 so
this is a little bit different it still

00:03:10.370 --> 00:03:15.890
looks a little bit bizarre 0 equals
0 last time we had 0 is equal to we

00:03:15.890 --> 00:03:20.780
add what 0 is equal to negative 20 now
we have 0 equals 0 so one way to think

00:03:20.780 --> 00:03:23.660
about it is even though the X's and
Y's are no longer in this equation

00:03:23.660 --> 00:03:28.190
okay well what XY pairs is it going to
be true for is that it's going to make

00:03:28.190 --> 00:03:32.420
it true that 0 is equal to 0 well this
is going to be true no matter what x and

00:03:32.420 --> 00:03:37.100
y are in fact x and y are not involved
in this equation anymore 0 is always

00:03:37.100 --> 00:03:41.870
going to be equal to 0 so this is going
to have an infinitely this is going to

00:03:41.870 --> 00:03:46.910
have infinitely many solutions here and
that's because these are the same lines

00:03:46.910 --> 00:03:50.390
they just look a little bit different
algebraically but if you scale one of

00:03:50.390 --> 00:03:54.320
them in the right way in fact if you
just multiply both sides of this one

00:03:54.320 --> 00:03:58.430
the second one by negative 2 you're
going to get the top one and so they

00:03:58.430 --> 00:04:01.910
actually represent the exact same
lines you have an infinitely many

00:04:01.910 --> 00:04:08.060
solutions all right when trying to
find the solution to the following

00:04:08.060 --> 00:04:12.320
system of linear equations Yvonne takes
several correct steps that lead to the

00:04:12.320 --> 00:04:16.970
equation negative 5 is equal to 20 how
many solutions is the system of linear

00:04:16.970 --> 00:04:19.640
equations have I don't even have to
look at the system right over here

00:04:19.640 --> 00:04:23.210
the fact that she got the statement
that can never be true negative 5 is

00:04:23.210 --> 00:04:26.930
never going to be equal to 20 tells
us that she has no those solutions

00:04:26.930 --> 00:04:29.930
and once again if you were to plot
these graphically you would see that

00:04:29.930 --> 00:04:33.230
these are parallel lines that's why they
have no solutions they never intersect

00:04:33.230 --> 00:04:39.590
there's no XY pair that satisfies both
of these constraints let's do a let's

00:04:39.590 --> 00:04:43.970
do a couple more of these when trying
to find the solution to the following

00:04:43.970 --> 00:04:47.180
system of linear equations Alba's take
several correct steps that lead to the

00:04:47.180 --> 00:04:51.320
equation 5y is equal to negative 5 and
say how many solutions does this system

00:04:51.320 --> 00:04:55.880
of linear equations have well 5y equals
negative 5 we could divide both sides

00:04:55.880 --> 00:05:01.370
by 5 and we get Y is equal to negative
1 and then if you substitute back in Y

00:05:01.370 --> 00:05:04.730
is equal to negative 1 if you did it
in this first equation if Y is equal

00:05:04.730 --> 00:05:09.800
to negative 1 all of this becomes
positive 2 you can subtract 2 from

00:05:09.800 --> 00:05:17.600
both sides and you get 5x is equal to
4 or you'd get what X is equal to 4/5

00:05:17.600 --> 00:05:23.930
or if you put negative 1 over here you
would get 5x minus 3 is equal to 1 you

00:05:23.930 --> 00:05:29.840
could add 3 to both sides and you get
5x equals 4 again X is equal to 4/5 so

00:05:29.840 --> 00:05:35.300
you have exactly one solution you
would have X is equal to 4/5 y is

00:05:35.300 --> 00:05:41.870
equal to negative 1 let's do one more
when trying to find the solution of the

00:05:41.870 --> 00:05:45.140
following system of linear equations
Levon take several correct steps that

00:05:45.140 --> 00:05:48.890
lead to the equation 0 equals 0 so once
again I don't even need to look at this

00:05:48.890 --> 00:05:52.970
over here zero equals zero is always
going to be true so this is going to

00:05:52.970 --> 00:05:57.770
have an infinitely this is going
to be an infinitely many solutions


Worked example: non-equivalent systems of equations

WEBVTT
Kind: captions
Language: en

00:00:00.307 --> 00:00:02.385
- Scarlett and Hansol's teacher gave them

00:00:02.385 --> 00:00:04.369
a system of linear equations to solve.

00:00:04.369 --> 00:00:05.922
They each took a few steps that lead

00:00:05.922 --> 00:00:07.825
to the systems shown in the table below.

00:00:07.825 --> 00:00:08.788
So this is the teacher system.

00:00:08.788 --> 00:00:11.308
This is what Scarlett got
after taking some steps.

00:00:11.308 --> 00:00:12.909
This is what Hansol got.

00:00:12.909 --> 00:00:14.106
Which of them obtained a system

00:00:14.106 --> 00:00:17.019
that is equivalent to
the teacher's system?

00:00:17.019 --> 00:00:17.681
And just to remind ourselves,

00:00:17.681 --> 00:00:19.701
an equivalent system is a system

00:00:19.701 --> 00:00:21.303
that has, or at least for our purposes,

00:00:21.303 --> 00:00:23.822
is a system that has the same solution,

00:00:23.822 --> 00:00:25.320
or the same solution set.

00:00:25.320 --> 00:00:26.898
So if there is a certain xy

00:00:26.898 --> 00:00:28.268
that satisfies this system

00:00:28.268 --> 00:00:31.160
in order for Scarlett's
system to be equivalent

00:00:31.160 --> 00:00:33.458
it needs to have the same solution.

00:00:33.458 --> 00:00:34.967
So let's look at this.

00:00:34.967 --> 00:00:36.406
So Scarlett, let's see,

00:00:36.406 --> 00:00:39.159
let's see if we can match these up.

00:00:39.159 --> 00:00:43.465
So her second equation here,

00:00:43.465 --> 00:00:45.473
so this is interesting,

00:00:45.473 --> 00:00:49.235
her second equation 14x - 7y = 2

00:00:49.235 --> 00:00:50.813
over here the teacher has an equation

00:00:50.813 --> 00:00:54.133
14x - 7y = 7.

00:00:54.133 --> 00:00:56.745
So this is interesting because the ratio

00:00:56.745 --> 00:00:59.334
between x and y is the same,

00:00:59.334 --> 00:01:01.354
but then your constant term,

00:01:01.354 --> 00:01:04.744
the constant term is
going to be different.

00:01:04.744 --> 00:01:06.114
And I would make the claim

00:01:06.114 --> 00:01:09.109
that this alone tells you
that Scarlett's system

00:01:09.109 --> 00:01:11.629
is not equivalent to the teacher.

00:01:11.629 --> 00:01:14.624
And you're saying, well,
how can I say that?

00:01:14.624 --> 00:01:16.690
Well, these two equations if you were

00:01:16.690 --> 00:01:19.128
to write them into slope intercept form,

00:01:19.128 --> 00:01:20.590
you would see because the ratio

00:01:20.590 --> 00:01:23.895
between x and y, the x
and y terms is the same.

00:01:23.895 --> 00:01:25.637
You're going to have the same slope,

00:01:25.637 --> 00:01:27.273
but you're going to have
different Y intercepts.

00:01:27.273 --> 00:01:28.690
In fact, we can actually solve for that.

00:01:28.690 --> 00:01:30.072
So this equation right over here

00:01:30.072 --> 00:01:32.927
we can write it as if we, let's see,

00:01:32.927 --> 00:01:35.133
if we subtract 14x from both sides

00:01:35.133 --> 00:01:38.761
you get -7y - 14,

00:01:38.761 --> 00:01:41.721
woops, -7y =,

00:01:41.721 --> 00:01:45.892
is equal to -14x + 7

00:01:45.892 --> 00:01:47.749
and we could divide both sides by -7.

00:01:47.749 --> 00:01:52.485
You get y = 2x -1 so that's this...

00:01:52.485 --> 00:01:54.321
All I did is algebraically
manipulate this.

00:01:54.321 --> 00:01:55.643
This is this line and I could even try

00:01:55.643 --> 00:01:58.046
to graph it so let's do that.

00:01:58.046 --> 00:02:00.484
So I'll draw a quick coordinate.

00:02:00.484 --> 00:02:02.087
This is just going to be very rough.

00:02:02.087 --> 00:02:04.084
Quick coordinate axis right over there,

00:02:04.084 --> 00:02:06.138
and then this line,

00:02:06.138 --> 00:02:08.901
this line would look something like this.

00:02:08.901 --> 00:02:12.457
So its y intercept is -1
and it has a slope of 2.

00:02:12.457 --> 00:02:16.196
So let me draw a line with a slope of,

00:02:16.196 --> 00:02:19.071
a line with a slope of 2 might look

00:02:19.071 --> 00:02:22.493
something like that.

00:02:22.493 --> 00:02:24.935
So that's this line right over here

00:02:24.935 --> 00:02:28.499
or this one right over there,

00:02:28.499 --> 00:02:29.753
and let's see.

00:02:29.753 --> 00:02:32.992
This one over here is going to be,

00:02:32.992 --> 00:02:35.058
if we do the same algebra,

00:02:35.058 --> 00:02:36.296
we're going to have

00:02:36.296 --> 00:02:41.156
- 7y = -14x + 2,

00:02:41.809 --> 00:02:45.028
or y = , I'm just
dividing everything by -7,

00:02:45.028 --> 00:02:49.423
2x - 2/7

00:02:50.173 --> 00:02:52.018
so this is going to look
something like this.

00:02:52.018 --> 00:02:56.522
Its y intercept is -2/7 so
it's like right over there.

00:02:56.522 --> 00:02:59.401
So this line is going
to look something like,

00:02:59.401 --> 00:03:00.377
I'm going to draw my best,

00:03:00.377 --> 00:03:02.791
my best attempt at drawing it,

00:03:02.791 --> 00:03:05.257
it's going to look something...

00:03:05.257 --> 00:03:07.294
Actually that's not quite right.

00:03:07.294 --> 00:03:08.462
It's going to look something like...

00:03:08.462 --> 00:03:10.551
I'll actually just start
it right over here.

00:03:10.551 --> 00:03:13.082
It's going to look something like,

00:03:13.082 --> 00:03:14.592
something like this.

00:03:14.592 --> 00:03:16.228
It's going to have the same slope,

00:03:16.228 --> 00:03:18.608
and obviously it goes in
this direction as well.

00:03:18.608 --> 00:03:20.686
Actually, let me just draw that.

00:03:20.686 --> 00:03:22.706
So it's going to have the same slope,

00:03:22.706 --> 00:03:25.643
but at different y intercepts.

00:03:25.643 --> 00:03:27.744
That doesn't look right,
but you get the idea.

00:03:27.744 --> 00:03:29.509
These two lines are parallel.

00:03:29.509 --> 00:03:30.507
So these two lines are parallel

00:03:30.507 --> 00:03:33.386
so any coordinate that satisfies this one

00:03:33.386 --> 00:03:34.942
is not going to satisfy this one.

00:03:34.942 --> 00:03:36.868
They have no points in common.

00:03:36.868 --> 00:03:37.786
They are parallel.

00:03:37.786 --> 00:03:40.166
That's the definition of parallel.

00:03:40.166 --> 00:03:43.567
Since this and this have
no points in common,

00:03:43.567 --> 00:03:45.448
there's no way that some solution set

00:03:45.448 --> 00:03:47.503
that satisfies this would satisfy this

00:03:47.503 --> 00:03:49.964
'cause any xy that satisfies this

00:03:49.964 --> 00:03:52.448
can't satisfy this or vice versa.

00:03:52.448 --> 00:03:53.249
They're parallel.

00:03:53.249 --> 00:03:54.364
There are no points.

00:03:54.364 --> 00:03:56.824
These two things will never intersect.

00:03:56.824 --> 00:04:00.006
So Scarlett does not have
an equivalent system.

00:04:00.006 --> 00:04:02.502
Now what about Hansol?

00:04:02.502 --> 00:04:04.104
Well, we see Hansol has the same thing

00:04:04.104 --> 00:04:05.346
going on here.

00:04:05.346 --> 00:04:09.688
5x - y, 5x - y, but then the constant term

00:04:09.688 --> 00:04:13.101
is different, -6, positive 3.

00:04:13.101 --> 00:04:16.502
So this and this also
represent parallel lines.

00:04:16.502 --> 00:04:19.091
Any xy pair that satisfies this,

00:04:19.091 --> 00:04:21.204
there's no way that it's
going to satisfy this.

00:04:21.204 --> 00:04:23.003
These two lines don't intersect.

00:04:23.003 --> 00:04:24.489
They are parallel.

00:04:24.489 --> 00:04:28.651
So Hansol's system is
not equivalent either.