# 7.4: Solving Systems of Equations by Multiplication

## Learning Objectives

At the end of this lesson, students will be able to:

- Solve a linear system by multiplying one equation.
- Solve a linear system of equations by multiplying both equations.
- Compare methods for solving linear systems.
- Solve real-world problems using linear systems by any method.

## Vocabulary

Terms introduced in this lesson:

- scalar
- lowest common multiple

## Teaching Strategies and Tips

Use the introduction to convince students that the elimination method applies to *any* linear system because one or both equations can be multiplied by a constant resulting in a “new” pair of equations with matching coefficients.

In Example 1, some students have trouble keeping track of the multipliers for each equation. Try using a visual:

In Example 2,

- Point out that the distance covered is the same in both directions, so the is unnecessary information.
- Remind students to back-substitute to complete the problem.

Use Example 3 to demonstrate a system with no matching coefficients and no coefficients that are multiples of others.

- Remind students how to find the LCM of two numbers.
- Suggest that students find the LCM of the “smaller pair” of coefficients, and , instead of and 1845.

Teachers may decide to forgo back-substitution and instead teach elimination of the second variable (“double elimination”).

Additional Example:

*Solve the system using multiplication.*

Solution by “double elimination”. Eliminate first.

Add.

Divide by . Therefore, .

To find , eliminate next.

Add.

Divide by . Therefore, .

Answer: The solution to the system is .

The advantage of “double elimination” is that the fraction does not need to be back-substituted.

## Error Troubleshooting

General Tip: Students forget to multiply *every* term in an equation by the scalar.

General Tip: Encourage students to eliminate the variable whose coefficients in both equations have the smallest LCM.

Remind students in *Review Problem* 2e to align the variables column-wise.