The Linear Combination Method

Step 1 Multiply one or both of the equations by a constant to obtain coefficients that differ only in sign for one of the variables

Step 2 Add the revised eqaution from Step 1. Combining like terms will eliminate one of the variables. Solve for the remaining variable.

Step 3 Substitute the value obtained in Step 2 into either of the original equations and solve for the other variable.

Equation One: 3x + 2y = 6
Equation Two: -6x - 3y = -6

1 Multiply the first equation by 2 so the x-coefficients differ only in sign.

Equation One: 3x + 2y = 6
Multiply by 2
Equation One: 6x + 4y = 12

2 Add the revised Equation One with Equation Two and solve for y.

Equation One: 6x + 4y = 12
Equation Two: -6x - 3y = -6
Simplify
y=6

3 Substitute the value of y into one of the original equations. Solve for x.

Equation One: 3x + 2y = 6
Substitute 6 for y
Equation One: 3x + 2(6) =6
Solve for x
x=-2

The solution is (-2,6).

Tickets for your school's football game are $3.00 for students and $5.00 for non-students. On Friday night 937 tickets are sold and $3943 is collected. How many tickets are sold to students? to non-students?

x=student tickets
y=non-student tickets

Equation One: 3x +5y = 3943
Equation Two: x + y + 937

Multiply the Second Equation by -3 so that the x-coefficients differ only by sign.

Equation Two: x + y = 937
Multiply by -3
Equation Two: -3x - 3y = -2811

Add the revised equation two with equation one and solve for y.

Equation Two: -3x - 3y = -2811
Equation One: 3x +5y = 3943
Simplify
y=566

Substitute the value of y into one of the original equations. Solve for x.

Equation Two: x + y = 937
Substitute 566 for y
Equation Two: x + 566 = 937
Solve for x
x=371

The solution is (371,566). Therefore 371 student tickets were sold and 566 non-student tickets were sold for the football game.