The Addition Method
The addition method can be used to eliminate any variable whose coefficients
are opposites. If neither variable has coefficients that are opposites, then
we use the multiplication property of equality to change the coefficients of the
variables, as shown in Examples 1 and 2.
Example 1
Using multiplication and addition
Solve the system by the addition method:
2x  3y = 13
5x  12y = 46
Solution
If we multiply both sides of the first equation by 4, the coefficients of y will be
12 and 12, and y will be eliminated by addition.
(4)(2x  3y) 
= (4)(13) 
Multiply each side by 4 
5x  12y 
= 46 

8x + 12y 
= 52 

5x  12y 
= 46 

Add 
3x 

= 6 

x 
= 2 

Replace x by 2 in one of the original equations to find y:
2x  3y 
= 13 
2(2)  3y 
= 13 
4  3y 
= 13 
3y 
= 9 
y 
= 3 
Because 2(2)  3(3) = 13 and 5(2)  12(3) = 46 are both true, the
solution set is {(2, 3)}.
Example 2
Multiplying both equations before adding
Solve the system by the addition method:
2x + 3y = 6
3x  5y = 11
Solution
To eliminate x, we multiply the first equation by 3 and the second by 2:
3(2x + 3y) 
= 3(6) 
Multiply each side by 3. 
2(3x  5y) 
= 2(11) 
Multiply each side by 2. 
6x + 9y 
= 18 

6x  10y 
= 22 

Add. 
y 
= 4 

y 
= 4 

Note that we could have eliminated y by multiplying by 5 and 3. Now insert y
= 4 into one of the original equations to find x:
2x + 3(4) 
= 6 
Let y = 4 in 2x + 3y = 6. 
2x + 12 
= 6 

2x 
= 6 

x 
= 3 

Check that (3, 4) satisfies both equations. The solution set is {(3, 4)}.
We can always use the addition method as long as the equations in a system are
in the same form.
