Explore the Lesson

A. Solving Systems of Linear Equations by Elimination

In the Warm Up, you explored another method of producing a new equation in one variable from a pair of equations in two variables. This strategy is often called elimination, and involves adding or subtracting entire equations. The following example outlines this strategy.

Elimination
a method for solving a system equations where two equations are added or subtracted to reduce the number of variables

 

Example 1

Solve the following system of equations by elimination.

Algebraic Solution Solution Strategy Modelled
with Scales and Weights

The system can be represented using weights on scales.

Subtracting the left side of one equation from the left side of another equation and subtracting the right side of one equation from the right side of another equation will produce a new equation.

There were an equal number of y-variables in the two equations, so the y-variable has been eliminated.

Solve the resulting equation to determine the value of x.

3 x plus 2 y is equal to 17. This means removing 3 x and 2 y from the left side of the first scale shown above and removing 17 from the right side of the first scale shown above will produce another balanced scale, as shown below.

 

Substitute this known value back into one of the original equations to determine y.

y = 1
The solution is (5, 1).

Verify the solution.


In Example 1, you may have noticed that the elimination strategy used a different method of reducing the number of variables than the substitution strategy, but once a single equation in one variable is produced, both methods are the same.

The elimination strategy worked in Example 1 because the y-coefficient was the same in both equations. When the two equations were subtracted, a new equation was produced that included only one variable, x.