The Addition Method
In this section we see another method for eliminating a variable in a system of
equations.
The Addition Method
In the addition method we eliminate a variable by adding the equations.
Example 1
An independent system solved by addition
Solve the system by the addition method:
3x 
 
5y 
= 9

4x 
+ 
5y 
= 23 
Solution
The addition property of equality allows us to add the same number to each side of
an equation. We can also use the addition property of equality to add the two left
sides and add the two right sides:
3x 
 
5y 
= 9 

4x 
+ 
5y 
= 23 

7x 


= 14 
Add. 


x 
= 2 

The yterm was eliminated when we added the equations because the coefficients of
the yterms were opposites. Now use x = 2 in one of the original equations to find y.
It does not matter which original equation we use. In this example we will use both
equations to see that we get the same y in either case.
3x  5y 
= 9 

4x + 5y 
= 23 
3(2)  5y 
= 9 
Replace x by 2. 
4(2) + 5y 
= 23 
6 5y 
= 9 
Solve fory y. 
8 + 5y 
= 23 
5y 
= 15 

5y 
= 15 
y 
= 3 

y 
= 3 
Because 3(2)  5(3) = 9 and 4(2) + 5(3) = 23 are both true, (2, 3) satisfies
both equations. The solution set is {(2, 3)}.
