GM Lesson 059 Variables on Both Sides

Learning Intentions

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

• Solve linear equations with variables on both sides.
• Collect like terms to isolate the variable.
• Explain each step as an equivalent equation.

Prerequisites

Students should already be able to:

• Solve one-step and two-step linear equations.
• Use inverse operations to isolate an unknown.
• Expand simple expressions such as .
• Collect like terms such as .
• Check solutions by substituting into the original equation.

Key Idea Summary

Some linear equations have the unknown variable on both sides.

For example:

To solve these equations, we use equivalent operations on both sides so that the equation remains balanced.

The main strategy is:

1. Move variable terms to one side.
2. Move number terms to the other side.
3. Divide to isolate the variable.
4. Check the solution by substitution.

An equivalent equation has the same solution as the original equation.

For example:

Subtracting from both sides gives:

This is equivalent because the same operation was applied to both sides.

Direct Instruction and Worked Examples

Time Allocation

Time Allocation

• Introduction, warmup and vocabulary: 5 minutes
• Direct instruction: 15 minutes
• Understanding checks: 5 minutes
• Exercises: 20 minutes
• Homework: 20 to 30 minutes outside the lesson it was taught in.

Link to original

Direct Instruction

When solving equations with variables on both sides, the goal is to create an equation with the variable on one side only.

For example:

Both sides contain an term. We can remove from the right side by subtracting from both sides.

This gives:

Now the equation is a familiar two-step equation.

The important principle is:

Worked Example 1: Variables on Both Sides

Solve:

Solution

Check:

Since both sides are equal, the solution is:

Worked Example 2: Choosing the Easier Side

Solve:

Solution

Move the smaller variable term, , so that the variable coefficient remains positive.

Check:

Therefore:

Worked Example 3: Variable Terms on Both Sides with Negative Coefficients

Solve:

Solution

Therefore:

It is acceptable to write the final answer as:

Worked Example 4: Expanding Before Solving

Solve:

Solution

First expand the bracket.

Therefore:

Worked Example 5: Practical Context

A gym charges a joining fee plus a weekly cost. Plan A costs $ to join and $ per week. Plan B costs $ to join and $ per week.

After how many weeks will the total cost be the same?

Solution

Let be the number of weeks.

Plan A:

Plan B:

Set the costs equal:

Therefore, the plans cost the same after:

Check:

Both plans cost $ after weeks.

Understanding Checks

Understanding Check 1

Solve:

Expected working:

Understanding Check 2

Explain why this step is valid:

Expected response:

Subtracting from both sides produces an equivalent equation. The equation remains balanced because the same operation is applied to both sides.

Understanding Check 3

A student solves:

Check whether the solution is correct.

Expected check:

The solution is correct.

Exercises

Simple Familiar Exercises

Exercise 1

Solve:

Exercise 2

Solve:

Exercise 3

Solve:

Exercise 4

Solve:

Exercise 5

Solve:

Exercise 6

Solve:

Exercise 7

Solve:

Exercise 8

Solve:

Complex Familiar Exercises

Exercise 9

Solve:

Exercise 10

Solve:

Exercise 11

Solve:

Exercise 12

Solve:

Exercise 13

Solve:

Exercise 14

Solve:

Exercise 15

A phone plan charges $ per month plus $ per text message. Another plan charges $ per month plus $ per text message.

Let be the number of text messages.

Write and solve an equation to find when the two plans cost the same.

Exercise 16

A taxi company charges a booking fee of $ plus $ per kilometre. Another company charges a booking fee of $ plus $ per kilometre.

Let be the distance in kilometres.

Write and solve an equation to find the distance where the two fares are equal.

Homework Problems

Homework Problem 1

Solve:

Homework Problem 2

Solve:

Homework Problem 3

Solve:

Homework Problem 4

Solve:

Homework Problem 5

Solve:

Homework Problem 6

Solve:

Homework Problem 7

A streaming service charges $ per month plus $ per movie rental. Another service charges $ per month plus $ per movie rental.

Let be the number of movie rentals.

Write an equation and solve it to find when the costs are equal.

Homework Problem 8

Check your answer to Homework Problem 7 by substituting the value of into both cost expressions. :)

Next: GM Lesson 060 Linear Equations with Fractions and Decimals