How to Solve Linear Systems by Multiplication

The multiplication method eliminates one variable from the linear system.

Elimination by multiplication is one of many methods of solving systems of linear equations. The goal of elimination is to cancel out one of the two variables, leaving a single equation with a single variable that can be solved by isolation. Multiply one or both of the equations in the system by a value that will make the coefficients of a variable equal. This will allow you to cancel out that variable when you subtract one of the equations from the other.

Instructions

1. Choose one of the variables to eliminate by multiplication in the system of equations. If possible, pick a variable with equal coefficients in the two equations or with one of the coefficients a multiple of the other. This will make the elimination easier to accomplish. For example, in the system of equations 2x - 3y = 6, x - 2y = 2, choose the variable x to eliminate.

2. Multiply both sides of one (or both, if necessary) of the equations by a number that will make the coefficient of the variable you chose in Step 1 the same in both equations. In the example, the coefficient of x is 2 in the first equation and 1 in the second equation, so multiply the second equation by 2 to make the coefficients of x equal: 2*(x - 2y) = 2*(2) yields the equation 2x - 4y = 4.

3. Subtract the second equation from the first equation by lining up the terms and subtracting term by term. This will cancel out the variable you chose to eliminate, leaving one equation with one variable. In the above example, subtracting the equation 2x - 4y = 4 from the equation 2x - 3y = 6 yields the equation -y = 2.

4. Solve for the value of the remaining variable in the equation from Step 3 using algebraic methods of isolation. In the example, isolate y in the equation -y = 2 by multiplying both sides of the equation by -1 to get the solution y = -2.

5. Substitute the value of this variable into one of the original equations in the system. For example, substitute -2 for y in the equation 2x - 3y = 6 to get the equation 2x + 6 = 6.

6. Solve for the value of the remaining variable in the equation from Step 5 using algebraic methods of isolation. In the example, isolate x by subtracting 6 from both sides of the equation to get 2x = 0, then dividing both sides of the equation by 2 to get the solution x = 0. The solution to the linear system is x = 0, y = -2, or (0, -2).

References

• Glencoe Mathematics Algebra 1: Elimination Using Multiplication