## Linear Equations: Solutions Using Elimination with Two Variables

To solve systems using elimination, follow this procedure.

• Arrange both equations in standard form, placing like variables and constants one above the other.
• 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.
• Add the equations, leaving one equation with one variable.
• Solve for the remaining variable.
• Substitute the value found in Step 4 into any equation involving both variables and solve for the other variable.
• Check the solution in both original equations.

Example 1

Solve this system of equations by using elimination.

Arrange both equations in standard form, placing like terms one above the other.

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.

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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