Linear Equations: Solutions Using Eliminations

To solve systems using elimination, follow this procedure.

1. Arrange both equations in standard form with like terms above one another.

2. Choose a variable to eliminate, and with a proper choice of multiplication, arrange so that the coefficients of that variable are opposites of one another.

3. Add the equations, leaving one equation with one variable.

4. Solve for the remaining variable.

5. Either repeat Steps 2 through 4, eliminating the other variable, or substitute the value found in Step 4 into any equation involving both variables and solve for the other variable.

6. Check the solution in both original equations.

Example 1: Solve this system of equations by using elimination.

Arrange both equations in standard form, putting like terms above one another.

Select a variable to eliminate, say y.

The coefficients of y are 5 and –2. These both divide into 10. Arrange so that the coefficient of y is 10 in one equation and –10 in the other. To do this, multiply the top equation by 2 and the bottom equation by 5.

Add the new equations, eliminating y.

Solve for the remaining variable.

Repeat Steps 2 through 4 (using new multiplication factors) to eliminate x and solve for y or substitute the above value for x and solve for y.

Method 1: Eliminate x and solve for y.

Method 2: Substitute for x and solve for y.

Check the solution in the original equation.

These are both true statements. The solution is .

If the elimination method produces a sentence that is always true, then the system is dependent, and either original equation is a solution. If the elimination method produces a sentence that is always false, then the system is inconsistent and there is no solution.

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