SOLVING LINEAR SYSTEMS WITH SUBSTITUTION

by Sam Callahan

By now you’ve learned to solve systems of equations using graphing and finding where the lines intersect:

73 yx

3xy

The problem with solving by graphing though, is evident when you look at graphs like the one below.

This solution (the blue point

where the lines intersect) isn’t

on a gridline and very

hard to accurately identify

3xy 73 yx

Although graphing is simple and visual, it is really only accurate enough to use with systems that have integer answers.

Substitution is a way to solve systems of equations analytically, or without graphing.

If we take the same system of equations:

We can make solving these equations possible by working with one variable at a time. To do this, we substitute one side of the equation in for the other variable.

3xy

73 yx

If we know that y = x+3, we can plug in (x+3) in for y in the second equation

to find out what x is.

3x – y = 7 substitute (x+3) for y

3x – (x+3) = 7 distribute -1 through the parentheses

3x – x – 3 = 7 simplify

2x – 3 = 7 simplify

2x = 10 simplify

x=5

x = 5

Now that we know what x is (5), we can plug 5 in for x in either equation to find out what y is. I like to use whichever equation has simpler numbers to work with.

In this case, that equation is: 3xy

If x = 5

y = x + 3y = 5 + 3

y = 8

x = 5 and y = 8, so our solution to the system

In (x, y) form is (5, 8)

Now let’s check our answer.

3xy

73 yx

Checking your solution

You should check your answer using the equation that you didn’t just solve.

For example, my last step was plugging in 5 for x into y = x + 3

I should check with the other equation, 3x – y = 7

Checking your solution

x = 5 y = 8

3x – y = 7 plug in your values for x and y

3(5) – (8) = 7 simplify

15 – 8 = 7 simplify

7 = 7 make sure your statement is true

We ended up with a true statement, so our solution works!

Try this one…

4x – 12y = 203x + 9y = 45

Try this one…

4x – 12y = 203x + 9y = 45

Unlike the previous example, we aren’t given an equation right away that says what x or y

is equal to, so we have to simplify one of these equations so that it reads

y=_____ or x=______

Choose one of the equations to simplify.I’ll use 3x + 9y = 45

3x + 9y = 45

You can start with either variable, but I want to solve for x first because I don’t want a fraction that would result if I divided everything by 9.

3x + 9y = 45 subtract 9y to put it on the right side of the equation

3x = 45 – 9y divide everything by 3 so that x will be on its own

/3 /3 /3

x = 15 – 3y

x = 15 – 3y

Now we have what x is equal to, so we can plug in“15 – 3y” for x in the other equation.

4x – 12y = 20 plug in the expression for x

4(15 – 3y) – 12y = 20 distribute

60 – 12y – 12y = 20 simplify

60 – 24y = 20 simplify

40 = 24y simplify

y = 5/3 I used a calculator for this step, but you could also simplify this fraction (40/24) by hand

y = 5/3

Plug this y-value into the other equation we found (x = 15 – 3y) to find x.

x = 15 – 3y plug in (5/3) for yx = 15 – 3(5/3) simplifyx = 15 – 5 3 times (5/3) is 5

x = 10

x = 10 y = 5/3

So our solution is (10, 5/3)

Always check your solution!

To review, 4x – 12y = 203x + 9y = 45

We simplified one of the equations3x + 9y = 45 x = 15 – 3y

Plugged this “15 – 3y” in for x in the other equation4x – 12y = 204(15 – 3y) – 12y = 20

Solved for yy = 5/8

Plugged in this y-value into the other equation to find xx = 15 – 3(5/3)x = 10