Elimination

Elimination Strategies

If substitution fails, another method to solve systems of equations is elimination.

The first step is to multiply one or both of the equations in such a way that the resulting coefficients on one variable have the same absolute value.

• One foolproof way to accomplish is to multiply one equation by the coefficient of the desired variable in the other equation, and do the same with the other equation.   
• Ex: 5x + 3y = 20 ; 3x + 10y = 30  -----> multiply 1st equation by 3, and the 2nd equation by 5 to eliminate x.

There are also certain shortcuts that allow you to only multiply one of the equations, such as if 3 and 6 are the coefficients of x, you could just double the 1st equation, or cut the 2nd in half.

• Ex: 2x + 3y = 7 ; 4x + 5y = 22 -------> simply multiple the 1st equation by 2.

Next, you must add/subtract the equations to eliminate the variable for which you made both variables have the same absolute value so that the variable is eliminated. Then, solve to find the other variable

• Ex: 2x + 3y = 8         

           +   -2x + 3y = 4
           =             6y = 12
                           y = 2

Then, substitute your solution (in this case, 2) into one of the original equations to find the value of the variable you eliminated.

• 2x + 3(2) = 8  ----> 2x = 2 -----> x=1

Finally, check your solutions by substituting both variables in the original equations.