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.