The addition method

In the addition method we eliminate a variable by adding the equations.

Example 1
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           add
7x         = 14
x = 2

The y-term was eliminated when we added the equations because the coefficients of the y-terms were opposites. Now use x = 2 in one of the original equations to find y:
3x – 5y = –9
3(2) – 5y = –9
6 – 5y = –9
–5y = –15
y = 3

Answer: The solution set is {(2, 3)}.

Example 2 Solve the system by the addition method:
2x – 3y = –13
5x – 12y = –46

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)
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

Answer: The solution set is {(–2, 3)}.


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