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# 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.

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