This document describes the addition/subtraction method for solving systems of linear equations algebraically. It involves eliminating one variable by adding or subtracting equations such that the coefficients of the variable you want to eliminate are the same or negatives of each other. Then the resulting single equation can be solved for the eliminated variable. The method is demonstrated by solving the systems (1) x - 2y = -9, x + 3y = 16 and (2) 2x - y = 9, 3x + 4y = -14, obtaining the solutions (1, 5) and (unknown).

1. Solving Linear Systems
Algebraically Using the
Addition/Subtraction Method
(aka the Elimination Method)

2. Using the Addition/Subtraction method is
another way to solve a system of linear
equations algebraically.
You can think of this method as temporarily
“eliminating” one of the variables to make your
life easier.
This method is another tool you can use to
solve simultaneous equations without graphing
them.

3. Follow along as the addition/subtraction
method is used to solve this system of
equations:
x - 2y = -9
x + 3y = 16

4.  First, be sure like variables are “lined
up” under one another. In this
x - 2y = -9
problem, they are already “lined up”. x + 3y = 16
 Decide which variable (x or y) will be
easier to eliminate. In order to
eliminate a variable, the numbers in
front of them (coefficients) must be
the same or negatives of one another.
 Looks like “x” will be the easier
variable to eliminate in this problem
since the x’s already have the same
coefficients (1).

5.  In this problem, we need to subtract to eliminate the “x” variable. Subtract all of the sets of
lined up terms. Remember: when you subtract signed numbers, you change the signs and
follow the rules for adding signed numbers.
x – 2y = -9
x + 3y = 16
becomes
x – 2y = -9
-x – 3 = -16
 Add vertically
x – 2y = -9
-x – 3y = -16
0 - 5y = -25
 Solve this simple equation
-5y = -25
y=5
 Substitute “ y = 5 ” into either of the original equations to get the value for “x”
x – 2(5) = -9
x – 10 = - 9
x= 1
 The solution is the ordered pair (1, 5)

6. You can check your 1-10 = -9
solution by -9 = -9
substituting x =1 and check!
y = 5 into both of the
original equations. If
the ordered pair is x + 3y = 16
correct, both 1 + 3(5)= 16
equations will be 1 + 15 = 16
true!
16 = 16
check!

7. Let’s try a more complicated problem.
Follow along as the addition/subtraction method is
used to solve this system of equations:
2x - y = 9
3x + 4y = -14