1. The Substitution Method
Find the solution.
2x – y = 13
– 4x – 9y = 7
This variable is the easiest to isolate.
y = 2x – 13
Solve the first equation for y.
– 4x – 9y = 7
This is the original second equation.
– 4x – 9(2x – 13) = 7
Substitute the expression into the other equation and solve.
– 4x – 18x = 7
– 22x = – 110
x = 5
Substitute this value into one of the original equations.
Continued 1

2. Example (cont) 2x – y = 13 – 4x – 9y = 7 2x – y = 13 2(5) – y = 13
First equation
Second equation
2x – y = 13
2(5) – y = 13
Substitute x = 5 into the first equation.
10 – y = 13
Solve for y.
–y = 3
y = –3
The solution is (5, –3).
Continued 2

3. Example (cont) Check the solution (5, –3) in both original equations.
2x – y = 13
– 4x – 9y = 7
2(5) – (3) = 13
– 4(5) – 9(3) = 7
= 13
– = 7
13 = 13 
7 = 7  3

4. The Addition Method Solve by addition. 5x – 3y = 14 2x – y = 6
Multiply each term of equation (2) by 3.
 6x + 3y = 18
This equation is equivalent to equation (2).
5x – 3y = 14
 6x + 3y = 18
 x =  4
Add the two equations.
Substitute this value into either equation to find y.
x = 4
Continued 4

5. Example (cont) 5x – 3y = 14 2x – y = 6 2x – y = 6 2(4) – y = 6
Substitute.
8  y = 6
y = 2
The solution is (4, 2).
y = 2 5

6. Example
Solve.
To eliminate the variable y, we want the coefficients of the y terms to be opposites. 6

7. Example The solution is (–1, 2) and it checks.
Multiply first equation by 3.
Multiply the second equation by 5.
Solve for x.
Replace x by –1 in equations (3), and solve for y.
The solution is (–1, 2) and it checks. 7