Using Substitution to Solve a System of Equations

Suppose we had a system of equations

2x + y = 4
and
3x + 2y = 5

How can we solve for x and y?

The good thing is that we have two equations for two variables.

One way is to solve on equation for y and substitute. Let us start with

2x + y = 4

subtract 2x from both sides

y = 4 - 2x we can use this informaiton to solve for x by substituting y into the other equation

3x + 2y = 5 becomes

3x + 2(4-2x) = 5

3x + 8 - 4x = 5

-x = -3
x=3

so x = 3

Now we can substitute x into our first equation to find y

2x + y = 4

6 + y = 4

y = -2

To check let us substitute our answers into the equations and see if we get the same answer:


2x + y = 4
and
3x + 2y = 5

x=3 y= -2

2(3) - 2 = 4
6-2 =4 Correct.

Next

3(3) + 2(-2) =5
9 - 4 = 5 Correct.

So our solutions to the system check OK and are correct.