In this section we see another method for eliminating a variable in a system of equations.

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 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. 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)}.