Main content

### Course: Algebra (all content) > Unit 5

Lesson 4: Solving systems of equations with substitution# Systems of equations with substitution: potato chips

To solve a system of equations using substitution...

1. Isolate one of the variables in one of the equations, e.g. rewrite 2x+y=3 as y=3-2x.

2. You can now express the isolated variable using the other one. *Substitute* that expression into the second equation, e.g. rewrite x+2y=5 as x+2(3-2x)=5.

3. Now you have an equation with one variable! Solve it, and use what you got to find the other variable.

## Want to join the conversation?

- What does he mean by the last video?(55 votes)
- Does Sal draw these pictures? if he does, I'm very impressed.(51 votes)
- I believe he does(9 votes)

- So, which method is faster/easier/recommended: elimination or substitution?(16 votes)
- It really depends on the situation. Sometimes it's more convenient to use one over the other.(20 votes)

- I don't know why my teacher put me into this class because i really don't know any of this. Every time i look at this question i get over whelmed because i dont get the problems he is doing.

😵😠🤯(15 votes)- Let's use this example:

{7x+10y=36

{2x-y=-9

First, label the equations. 7x+10y=36 would be Equation 1, and 2x-y=-9 would be Equation 2. Next, multiply Equation 2 by an integer so that either the x values (2x and 7x) or the y values (-y and 10y) are equal. You could multiply Equation 2 by -10, for example, so the y values would be equal, since -y • -10 = 10y. Label the new equation, -20x+10y=90, Equation 3. Next, subtract Equation 3 from Equation 1. You should get 27x = -54. Divide both sides of the equation by 27, and you should get x=-2, so you've solved for x. Now, substitute x=-2 into Equation 2. You should get (-4)-y=(-9) (it is okay if you don't have the parentheses, I just put them there to emphasize the fact that -4 and -9 are negative). Add 4 to both sides of the equation to get - y=-5. Divide both sides of the equation by -1 to get y=5. You are done! (x,y)=(-2,5)!

My explanation is a little complicated, you might have to reread this. Hope it helps!(17 votes)

- the value for m = -4w+ 11 @1:49essentially the equation for the line which you could then use to find the answer to any combination/ variable to the problem(10 votes)
- This does not make any sense to me at all, could you explain this in easier terms?(6 votes)
- I can try, though I'm not sure if this is easier or not. A system of equations is where you have more than one equation with the same variables and you need to find out what values of the variables will work for all the equations.

Here is an example: 2x+3y=12; 5x+y=17

Substitution is one way to solve it.

First, we can rearrange one of the equations in order to isolate one of the variables:

5x+y=17

y=17-5x

We now have a way to express y in terms of x, so we can put it into the other equation instead of y in order to solve for x:

2x+3y=12

2x+3(17-5x)=12

2x+51-15x=12

51-13x=12

-13x=-39

x=3

Now we know what value x needs to be to satisfy both equations, so we can use that in place of x to solve for y.

5x+y=17

5(3)+y=17

15+y=17

y=2

We now know the values for both x and y, and in order to be quite sure, we can check them by putting them both into the first equation again:

2(3)+3(2)?=12

6+6?=12

12=12

This system of equations could also be solved in several other ways, such as elimination or graphing. In graphing, the point at which all equations in the system intersect is the solution.(6 votes)

- Why does and how does this substitution work?(5 votes)
- because "m" is always equal to the same thing. "M' will always equal -4w + 11, no matter which equation it's in(7 votes)

- Im still confused on everything..(6 votes)
- Would you like my help? Reply if you would like my help!(4 votes)

- How do we solve y intercept?(6 votes)
- I believe the elimination method is the easiest, but when would the substitution method be more useful?(3 votes)
- When one has messy equations that cannot be ''scaled'' easily in such a way that allows you to make an elimination. For instance is you have these 2 equations:

ln(3x + 2y) + 3*log(x/3) = 5

(x - 2)²/9 + (y + 7 )²/16 = 17x

It seems difficult to try making one of the sides of one of them equal to one side of the other equation. Painfully, substitution is the best choice to combine them.

Another part where substitution is useful is when you start to deal with composition of functions (videos on the Algebra II playlist talk about it), which is an important aspect when you start to learn calculus.(6 votes)

## Video transcript

Just as you were
solving the potato chip conundrum in the last video,
the king's favorite magical bird comes flying along and starts
whispering into the king's ear. And this makes you a
little bit self-conscious, a little bit
insecure, so you tell the king, what is the
bird talking about. And the king says,
well, the bird says that he thinks that there's
another way to do the problem. And you're not used to
taking advice from birds. And so being a
little bit defensive, you say, well, if the bird
thinks he knows so much, let him do this problem. And so the bird whispers
a little bit more in the king's ear and
says, OK, well I'll have to do the writing
because the bird does not have any hands, or at least
can't manipulate chalk. And so the bird continues to
whisper in the king's ear. And the king
translates and says, well, the bird says, let's
use one of these equations to solve for a variable. So let's say, let's us this
blue equation right over here to solve for a variable. And that's essentially
going to be a constraint of one variable
in terms of another. So let's see if we can do that. So here, if we want
to solve for m, we could subtract 400
w from both sides. And we would have 100 m. If we subtract 400w from the
left, this 400w goes away. If we subtract 400w
from the right, we have is equal to
negative 400w plus 1,100. So what got us from
here to here is just subtracting 400w
from both sides. And then if we want
to solve for m, we just divide
both sides by 100. So we just divide all
of the terms by 100. And then we get m is equal to
negative 400 divided by 100, is negative 4w. 1,100 divided by 100 is 11. Plus 11. So now we've constrained
m in terms of w. This is what the
bird is saying, using the king as his translator. Why don't we take
this constraint and substitute it back for
m in the first equation? And then we will have one
equation with one unknown. And so the king starts to
write at the bird's direction. 200, so he's looking at
that first equation now, he says 200. Instead of putting
an m there, the bird says well, by the
second constraint, m is equal to
negative 4w plus 11. So instead of writing an
m, we substitute for m the expression
negative 4w plus 11. And then we have the rest of it,
plus 300w, is equal to 1,200. So just to be clear,
everywhere we saw an m, we replaced it with
this right over here, in that first equation. So the first thing, you
start to scratch your head. And you say, is this a
legitimate thing to do. Will I get the same
answer as I got when I solved the same
problem with elimination? And I want you to sit and
think about that for a second. But then the bird starts
whispering in the king's ear. And the king just progresses to
just work through the algebra. This is one equation
with one unknown now. So the first step would
be to distribute the 200. So 200 times negative
4w is negative 800w. 200 times 11 is 2,200. Plus 2,200. And then we have the plus 300w. Plus 300w is equal
to positive 1,200. Now we just need to solve for w. We first might want to
group this negative 800w with this 300w. Negative 800 of something
plus 300 of something is going to be negative 500w. And then we still have this
plus 2,200 is equal to 1,200. Now to solve for w, we'd want to
subtract 2,200 from both sides. So subtract 2,200,
subtract 2,200. On the left-hand
side, you're left just with the negative 500w. And on the right-hand side, you
are left with negative 1,000. This is starting to
look interesting, because if we divide both
sides by negative 500, we get w is equal to 2, which
is the exact same answer that we got when we tried to figure
out how many bags of chips each woman would eat on average. When we tried to solve
it with elimination, we got the exact right answer. So at least for this example,
it seems like the substitution method that this bird came
up with worked just as well as the elimination method
that you had originally done the first time you wanted
to figure out the potato chip conundrum. And if now, you actually
wanted to figure out how many chips
the men would eat, well, you could do
exactly the same thing you did the last time. You know one of the variables. You can substitute it back
into one of the equations and then solve for m. And you could try that yourself
to verify that you actually will get the same
value for m as well. And in fact, this would
probably be the easiest equation to substitute into,
because it explicitly solves for m already.