Solve this linear system of equations by the substitution method.

-x + y = 3

2x – 2y = -6

To use the substitution method, we need something to substitute. That's kind of obvious; why did we feel the need to point it out? Anyway, let's isolate y from the first equation.

y = x + 3

Now stick it in the second equation.

2x – 2(x + 3) = -6

(2x – 2x) – 6 = -6

0 = 0

Zero always equals itself, so we have an infinite number of solutions. Every point on one line is also on the other.

Solve this linear system of equations by the substitution method.

x + 4y = 16

3x – 2y = 6

Solving the first equation for x looks like a winner.

x = -4y + 16

Now plug this into the second equation. The substitution method: the turducken of math.

3x – 2y = 6

3(-4y + 16) – 2y = 6

-12y + 48 – 2y = 6

-14y = -42

y = 3

Plug this into either of the original given equations. The first equation has treated us well so far, so we'll keep rolling with those good vibes.

x + 4y = 16

x + 4(3) = 16

x + 12 = 16

x = 4

The solution is (4, 3). Let's check our answer in the other original equation, 3x – 2y = 6.

Does 3(4) – 2(3) = 6?

12 – 6 = 6, so yeah, it does.

Solve the system of equations using the substation method.

To use the substitution method, we should solve one equation for one variable. Let’s solve our first equation for x. A little rearranging and we have .

Use the old switcheroo to put  in for x in the second equation:

Zero has a tendency to only equal zero, so we think this is false. So, there are no solutions to this system of equations. They're parallel lines that never meet. Sad, isn't it?

Solve this linear system of equations by the substitution method.

y = -x + 1

Both equations equal y already, so we can set them equal to each other. Mighty convenient, we say.

Multiplying everything by 2 clears the fraction from the right side.

-2x + 2 = -x + 8

-2x + x = 8 – 2

-x = 6

x = -6

Plug x = -6 into the second equation, y = -x + 1. Y'know, the one without a fraction.

y = -(-6) + 1

y = 7

We've found (-6, 7) as our solution. We can check our answer in the other original equation, .

Does ?

7 = 3 + 4?

It does. That's mighty convenient, too.