Solving a system of equations by substitution

The other day, we talked about solving a system of equations by graphing. Today, we’re going to look at solving a system of equations by substituting.

As the name suggests, this method involves substituting. This involves solving one equation for either x or y, and then placing that equation into the other to solve for the other variable. This is a bit labor-intensive, but it’s worth it since many find this easier than using elimination to solve a system.

Let’s solve this system of equations by substitution:

x + y = 12

2x + 3y = 9

We’re going to start by solving the first equation for y. We do this by subtracting x from both sides of the equation, leaving us with y = 12 – x.

Now, we’re going to substitute 12 – x everywhere we see y in the second equation. We can then solve for x. The whole process looks like this.

2x + 3(12 – x) = 9

2x + 36 – 3x = 9

2x – 3x + 36 = 9

-x + 36 = 9

-x = -25

x = 25

We now know the value for x, so we can substitute that in for the x in either equation. For this example, I’m going to substitute it back into the first equations.

25 + y = 12

y = -13

The one point that lies on both lines, therefore, is (25, -13).

We have one more method, elimination. Look for that one later this week!