🚀 StudyNest is growing. New lessons and worksheets are added regularly.

📘 Algebra • Lesson 10 of 22

Word Problems with Linear Equations

Word problems are real-life situations written using sentences. In this lesson, you will learn how to turn those sentences into equations and solve them step by step.

✏️ Algebra 📘 Worked examples included 🧠 Step-by-step learning

Learning Objectives

You should be able to:

Introduction

This lesson develops Word Problems with Linear Equations from meaning and patterns before moving to calculations. Look for what the quantities, shapes or symbols represent, then connect that idea to the method.

Key Notes

Read the definitions and key facts below carefully. Each rule is most useful when you can explain why it fits the situation, not simply recall it.

🎬 Translate words into algebra

A number plus 7 is 15
→
x + 7 = 15

First choose the unknown, then translate the relationship.

🤔 Think About This...

Your mother gives you some money.

Later, your grandmother gives you another $10.

You now have $45 altogether.

How much money did your mother give you?

We do not know the starting amount, so let:

x = the amount your mother gave you

The story can now be written as:

x + 10 = 45

Subtract 10 from both sides:

x = 35
Your mother gave you $35.

The words looked long, but the equation made the problem much easier.

📖 What is a Word Problem?

A word problem is a mathematical problem presented as a real-life situation using words.

Your first job is not to calculate immediately.

Your first job is to understand what the story is telling you and turn it into an equation.

Once you have written the correct equation, you can use the skills from Lesson 9 to solve it.

🧠 The StudyNest Thinking Method

Read → Represent → Solve → Answer
  1. Read: Read the problem carefully and decide what you are trying to find.
  2. Represent: Choose a letter for the unknown and write an equation.
  3. Solve: Solve the equation using inverse operations.
  4. Answer: Write a complete answer using the correct units or description.
Do not rush into calculations. First turn the story into Mathematics.

🔍 Useful Words and Their Meanings

Certain words can help you decide which operation to use.

Words Possible Operation
more than, increased by, altogether, total Addition (+)
less than, decreased by, difference, remaining Subtraction (−)
times, groups of, multiplied by Multiplication (×)
shared equally, divided by, per group Division (÷)
is, gives, equals, has a total of Equals sign (=)
These words are clues, but always read the whole sentence before choosing an operation.

📝 Worked Examples

Example 1: Sweets

Mary had some sweets. She bought 8 more and then had 25 sweets.

How many sweets did she have at first?

🔍 StudyNest Question: What are we trying to find?

We are trying to find the number of sweets Mary had at first.

Let x = the number of sweets she had at first

Write the equation:

x + 8 = 25

Solve:

x + 8 − 8 = 25 − 8
x = 17
Mary had 17 sweets at first.

Example 2: A Number

Six times a number is 42.

Find the number.

Let x = the unknown number

Six times the number means:

6x

Write the equation:

6x = 42

Solve:

x = 42 ÷ 6
x = 7
The number is 7.

Example 3: Ages

Tendai is 5 years older than his brother.

Tendai is 18 years old.

How old is his brother?

Let x = the brother's age

Tendai's age is the brother's age plus 5:

x + 5 = 18

Solve:

x = 18 − 5
x = 13
Tendai's brother is 13 years old.

Example 4: Kombi Fare

Five identical kombi trips cost $15 altogether.

Find the cost of one trip.

Let x = the cost of one trip

Write the equation:

5x = 15

Solve:

x = 15 ÷ 5
x = 3
One kombi trip costs $3.

🌍 More Real-Life Examples

🏪 Shopping

A loaf of bread costs x dollars. A bottle of milk costs $3. Together they cost $8.

x + 3 = 8

🎒 School Fees

Your parents have already paid $120. The total school fees are $350. How much is still owed?

120 + x = 350

🚰 Water Bottles

Four identical bottles cost $20. Find the cost of one bottle.

4x = 20

⚠ Common Mistakes

Mistake 1

Solving before writing an equation.

Always write the equation first.

Mistake 2

Forgetting to explain what the variable represents.

Let x = ...
Mistake 3

Giving only the value of x without answering the question.

❌ x = 13

✅ Tendai's brother is 13 years old.

Practice Questions

Write an equation first, then solve it.

  1. Sarah had some money. She received $15. She now has $40. How much money did she have at first?
  2. Eight times a number is 56. Find the number.
  3. A father is 30 years older than his son. The father is 52 years old. Find the son's age.
  4. Five identical pens cost $25. Find the cost of one pen.
  5. A book costs x dollars. Together with a ruler costing $4, the total is $15. Find the price of the book.
✅ Show Answers
  1. x + 15 = 40, therefore x = 25.
  2. 8x = 56, therefore x = 7.
  3. x + 30 = 52, therefore x = 22 years.
  4. 5x = 25, therefore x = $5.
  5. x + 4 = 15, therefore x = $11.

🧠 Did You Know?

Every engineer, accountant, scientist and programmer regularly solves word problems. The first step is almost always turning words into an equation.

⭐ What You Can Do Now

🎉 Well Done!

You have completed another important Algebra skill. Word problems become much easier when you follow the same process every time.

StudyNest Method:

📖 Read → ✏️ Represent → 🧮 Solve → ✅ Answer

🧠 Let’s Think Together

Before calculating, identify the information given, the result required and the mathematical relationship that connects them. Predict whether the answer should be larger, smaller or unchanged, then use that prediction to check the result.

📈 Your Progress

Lesson 10 of 22

Course Progress 63%

Exam Focus

Show the mathematical relationship you are using, keep notation consistent and give a clear final answer. Use a quick estimate, diagram or inverse operation to check that the result is reasonable.

Summary