Explore the Lesson

A. Solving Systems of Linear Equations by Substitution

In the Warm Up, a balance analogy was used to introduce a strategy for solving a system of equations that involved substituting one expression for another equivalent expression. This strategy is often called substitution. The goal of this type of substitution is to reduce two equations in two variables to a single equation in one variable, and then to solve the equation in one variable. The following example shows a solution strategy and a corresponding diagram.

Substitution
A method for solving a system equations where a variable's equivalent expression is substituted for that variable in another equation to reduce the number of variables.

 

Example 1

Solve the following system of equations.


Algebraic Solution Solution Strategy Modelled
with Scales and Weights

The system can be represented using weights on scales.

Looking at the first equation, you can see that x and 3 y are equal. This means you can replace the x in the second equation with 3 y.

Notice that the original equation had two variables, but the new equation only has one, y. Simplify and solve this new equation.

Now the y-value is known.

To complete the solution, substitute the y-value back into either one of the original equations to determine the x-value.

The solution is (6, 2).

Substitute these x- and y-values into each of the original equations to verify the solution.