GM Lesson 061 Equations from Words

Learning Intentions

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

• Translate descriptions in words into linear equations.
• Identify the unknown quantity in a worded situation.
• Develop equations that represent practical relationships.

Prerequisites

Students should already be able to:

• Solve one-step and two-step linear equations.
• Solve equations with variables on both sides.
• Solve equations involving fractions and decimals.
• Use inverse operations to isolate an unknown.
• Check a solution by substituting it back into the original equation.

Key Idea Summary

A worded situation can often be represented using a linear equation.

The first step is to define the unknown quantity clearly.

For example:

Let be the number of movie tickets bought.

Then use the words in the problem to build an equation.

Common word translations include:

Words	Algebra
a number
more than a number
less than a number
twice a number
half a number
the total is
the same as

A good equation should match the situation, not just contain the numbers from the question.

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

Warm-up

Translate each phrase into algebra.

1. more than a number
2. times a number
3. less than twice a number
4. Half a number is
5. The sum of a number and is

Direct Instruction

When forming an equation from words, use the following process.

1. Identify the unknown.
2. Define a variable.
3. Translate the words into algebra.
4. Form an equation.
5. Solve the equation if required.
6. Interpret the answer in context.

A linear equation from words usually represents a situation where an unknown quantity is being added, subtracted, multiplied, divided, or compared to another quantity.

Worked Example 1: A Number Problem

A number increased by is . Form and solve an equation to find the number.

Let be the number.

The phrase “increased by ” means .

So the equation is:

Solve:

Therefore, the number is .

Worked Example 2: Twice a Number

Twice a number is more than . Form and solve an equation.

Let be the number.

Twice the number is .

more than is .

So:

Solve:

Therefore, the number is .

Worked Example 3: Practical Cost Situation

A taxi charges a fixed booking fee of $ plus $ per kilometre. A trip costs $ . Form and solve an equation to find the distance travelled.

Let be the distance travelled in kilometres.

The total cost is:

The total cost is $ , so:

Solve:

Therefore, the taxi travelled km.

Worked Example 4: Variables on Both Sides

A gym has two payment options.

Option A charges $ per visit.

Option B charges a joining fee of $ plus $ per visit.

Form and solve an equation to find when the two options cost the same.

Let be the number of visits.

Option A costs:

Option B costs:

The costs are the same when:

Solve:

Therefore, the two options cost the same after visits.

Worked Example 5: Consecutive Numbers

The sum of two consecutive whole numbers is . Form and solve an equation to find the numbers.

Let be the smaller number.

The next consecutive whole number is .

Their sum is , so:

Solve:

The smaller number is .

The larger number is:

Therefore, the two numbers are and .

Understanding Checks

Check 1

Translate each phrase into algebra.

1. more than a number
2. less than a number
3. times a number
4. A number divided by
5. more than twice a number

Check 2

For each situation, define the unknown and write an equation. Do not solve yet.

1. A number increased by is .
2. Three times a number is .
3. A number divided by is .
4. The cost of tickets at $ each is $ .
5. A phone plan costs $ plus $ per message. The total cost is $ .

Check 3

A student writes the following equation for the situation:

“Five more than twice a number is .”

Student’s equation:

Explain why this equation is incorrect, then write the correct equation.

Expected correction:

Exercises

Simple Familiar Exercises

Exercise 1

Translate each phrase into algebra.

1. more than a number
2. less than a number
3. times a number
4. Half a number
5. more than times a number
6. less than twice a number

Exercise 2

For each statement, write a linear equation.

1. A number plus is .
2. A number minus is .
3. Twice a number is .
4. Three times a number is less than .
5. Half a number is .
6. more than a number is .

Exercise 3

Define the unknown and write an equation for each situation.

1. A movie ticket costs $ . The total cost is $ .
2. A student saves $ each week. After some weeks, they have saved $ .
3. A plumber charges $ call-out fee plus $ per hour. The total cost is $ .
4. A gym charges $ per month. The total cost after some months is $ .
5. A delivery company charges $ plus $ per kilometre. The total cost is $ .

Exercise 4

Write and solve an equation for each number problem.

1. A number plus is .
2. A number decreased by is .
3. Four times a number is .
4. more than twice a number is .
5. Three times a number minus is .

Complex Familiar Exercises

Exercise 5

A phone plan charges a fixed monthly fee of $ plus $ per text message. The monthly bill is $ .

1. Define the unknown.
2. Write a linear equation.
3. Solve the equation.
4. Interpret the answer in context.

Exercise 6

A car hire company charges $ per day plus a one-off booking fee of $ . A customer pays $ .

1. Let be the number of days.
2. Write an equation for the total cost.
3. Solve the equation.
4. State the number of days the car was hired.

Exercise 7

A rectangle has a length that is cm longer than its width. The perimeter is cm.

1. Let be the width.
2. Write an expression for the length.
3. Write an equation using the perimeter.
4. Solve to find the width and length.

Exercise 8

A school concert sells adult tickets for $ each and student tickets for $ each. A family buys adult tickets and some student tickets. The total cost is $ .

1. Define the unknown.
2. Write a linear equation.
3. Solve the equation.
4. Interpret the result.

Exercise 9

Two payment options are available for a streaming service.

Plan A costs $ per month.

Plan B costs a joining fee of $ plus $ per month.

1. Let be the number of months.
2. Write an expression for the cost of Plan A.
3. Write an expression for the cost of Plan B.
4. Form an equation for when the plans cost the same.
5. Solve the equation and interpret the result.

Exercise 10

The sum of three consecutive whole numbers is .

1. Let be the smallest number.
2. Write expressions for the next two numbers.
3. Form a linear equation.
4. Solve the equation.
5. State the three numbers.

Homework Problems

Problem 1

Translate each phrase into algebra.

1. more than a number
2. less than a number
3. times a number
4. One quarter of a number
5. more than twice a number

Problem 2

Write an equation for each statement.

1. A number plus is .
2. A number decreased by is .
3. Twice a number is .
4. Four times a number plus is .
5. Half a number plus is .

Problem 3

A gardener charges $ per hour plus a fixed travel fee of $ . The total cost is $ .

1. Let be the number of hours worked.
2. Write an equation.
3. Solve the equation.
4. Interpret the answer in context.

Problem 4

A rectangle has width cm and length cm. Its perimeter is cm.

1. Write an equation for the perimeter.
2. Solve for .
3. State the width and length.

Problem 5

A number is multiplied by , then is subtracted. The result is .

1. Define the unknown.
2. Write an equation.
3. Solve the equation.
4. Check the solution by substitution.

Next: GM Lesson 062 Practical Problems Involving Linear Equations