Linear Equation
Word Problems

Learn to translate real-world situations into equations, solve for unknowns, and interpret your answers in context.

Pre-Algebra Algebra

What Is a Linear Equation?

A linear equation is any equation that can be written in the form:

The variable $ x $ appears only to the first power: no $ x^2 $, no $ \sqrt{x} $, no $ \frac{1}{x} $. The graph of every linear equation is a straight line.

Why Word Problems Matter

Real life doesn't hand you equations. It gives you sentences. Word problems teach you to:

Translate

Turn English sentences into mathematical equations

Solve

Use algebra to isolate the unknown variable

Interpret

Check that your answer makes sense in the real world

Key Vocabulary to Know

Variable

A letter (like $ x $ or $ n $) that represents an unknown quantity

Coefficient

The number multiplied by the variable, e.g., the 3 in $ 3x $

Constant

A fixed number that doesn't change, like the 5 in $ x + 5 = 12 $

Expression vs. Equation

An expression has no $ = $ sign; an equation always has one

Translating Words to Math

These key phrases tell you which operation to use:

English Phrase	Math Symbol	Example
sum of, more than, increased by	$ + $	$ x + 7 $
difference, less than, decreased by	$ - $	$ x - 3 $
product of, times, of	$ \times $	$ 4x $
quotient of, divided by, per	$ \div $	$ \frac{x}{5} $
is, equals, results in	$ = $	$ x + 7 = 15 $

The Core Framework

Most linear word problems follow this common pattern:

The rate is how fast things change, $ b $ is where you start, and the total is what you end up with.

Expression vs. Equation

Students often confuse these two. Here is the critical difference:

	Expression	Equation
Has $ = $ sign?	No	Yes
Example	$ 3x + 5 $	$ 3x + 5 = 20 $
Can you solve for $ x $?	No: you can only simplify	Yes: find a specific value
Word-problem role	Describes one side	Connects two sides to find the unknown

The 5-Step Method

1

Read carefully. Identify what is known and what is unknown.

2

Define the variable. Write "Let $ x = $..." in a full sentence.

3

Write the equation. Translate phrases into math using the keyword table.

4

Solve. Use inverse operations to isolate $ x $.

5

Check and interpret. Plug back in: does the answer make sense in context?

Example 1: Movie Tickets

1

Define: Let $ x $ = number of tickets

2

Equation: $ 9x + 5 = 50 $

3

Solve: $ 9x = 45 $ → $ x = 5 $

4

Check: $ 9(5) + 5 = 45 + 5 = 50 $ ✓

Visualizing the Movie Ticket Problem

The bar model shows how the $5 fee plus five $9 tickets add up to the $50 total.

Example 2: Saving Money

1

Define: Let $ h $ = hours of babysitting

2

Equation: $ 12h + 30 = 126 $

3

Solve: $ 12h = 96 $ → $ h = 8 $

4

Check: $ 12(8) + 30 = 96 + 30 = 126 $ ✓

Example 3: Two Unknowns, One Equation

1

Define: Let $ w $ = width. Then length = $ w + 3 $.

2

Perimeter formula: $ 2(w + 3) + 2w = 46 $

3

Solve: $ 2w + 6 + 2w = 46 $ → $ 4w = 40 $ → $ w = 10 $

4

Check: Width = 10, Length = 13. Perimeter: $ 2(13) + 2(10) = 46 $ ✓

The Key Insight in Example 3

Even though the problem has two unknowns (width and length), the phrase "3 cm more than" creates a relationship between them:

This lets us write everything in terms of one variable, turning a two-unknown problem into a single linear equation. Watch for relationship phrases:

"more than"

$ w + k $

"twice as many"

$ 2w $

"less than"

$ w - k $

Common Mistakes to Avoid

✕ Reversing "Less Than"

"5 less than $ x $" is $ x - 5 $, not $ 5 - x $. The order matters.

✕ Forgetting Units

Writing just "8" instead of "8 hours". Always include the unit in your final answer.

✕ Not Checking

Plugging back in takes 10 seconds and catches sign errors, dropped terms, and misread problems.

Summary: Problem Types at a Glance

Problem Type	Equation Pattern	Watch For
Cost / Purchase	$ px + f = T $	Fixed fee + per-item cost
Saving / Earning	$ rx + s = G $	Starting amount + rate
Geometry (perimeter)	$ 2l + 2w = P $	Relationships between sides
Age / Comparison	$ x + (x + d) = S $	"older than", "twice as old"
Distance / Rate	$ rt = d $	Speed, time, distance triangle

Question 1 of 5

A gym charges a $25 registration fee plus $15 per month. If Carlos paid $145 total, which equation finds the number of months $ m $?

The monthly charge $ 15 $ is the rate (coefficient of $ m $), and the one-time registration fee $ 25 $ is the constant. Together they equal the total: $ 15m + 25 = 145 $.

Question 2 of 5

Solve: A number increased by 7 is 22. What is the number?

"Increased by 7" means $ x + 7 $, and "is 22" means $ = 22 $. So $ x + 7 = 22 $, giving $ x = 15 $. Check: $ 15 + 7 = 22 $ ✓

Question 3 of 5

Emma earns $8 per hour. After working some hours, she has $134. She started with $30. How many hours did she work?

Set up the equation: $ 8h + 30 = 134 $. Subtract 30: $ 8h = 104 $. Divide by 8: $ h = 13 $. Check: $ 8(13) + 30 = 104 + 30 = 134 $ ✓

Question 4 of 5

Which phrase translates to $ x - 9 $?

"9 less than a number" means you start with the number and subtract 9: $ x - 9 $. Be careful: "9 less than $ x $" is not $ 9 - x $.

Question 5 of 5

The perimeter of a triangle is 42 cm. The three sides are $ x $, $ x + 5 $, and $ 2x - 3 $. What is $ x $?

Add all three sides: $ x + (x + 5) + (2x - 3) = 42 $. Combine like terms: $ 4x + 2 = 42 $. So $ 4x = 40 $, giving $ x = 10 $. Sides are 10, 15, and 17 cm. Check: $ 10 + 15 + 17 = 42 $ ✓

English Phrase	Math Symbol	Example
sum of, more than, increased by	\( + \)	\( x + 7 \)
difference, less than, decreased by	\( - \)	\( x - 3 \)
product of, times, of	\( \times \)	\( 4x \)
quotient of, divided by, per	\( \div \)	\( \frac{x}{5} \)
is, equals, results in	\( = \)	\( x + 7 = 15 \)

	Expression	Equation
Has \( = \) sign?	No	Yes
Example	\( 3x + 5 \)	\( 3x + 5 = 20 \)
Can you solve for \( x \)?	No: you can only simplify	Yes: find a specific value
Word-problem role	Describes one side	Connects two sides to find the unknown

Problem Type	Equation Pattern	Watch For
Cost / Purchase	\( px + f = T \)	Fixed fee + per-item cost
Saving / Earning	\( rx + s = G \)	Starting amount + rate
Geometry (perimeter)	\( 2l + 2w = P \)	Relationships between sides
Age / Comparison	\( x + (x + d) = S \)	"older than", "twice as old"
Distance / Rate	\( rt = d \)	Speed, time, distance triangle

Linear EquationWord Problems

What Is a Linear Equation?

Why Word Problems Matter

Key Vocabulary to Know

Translating Words to Math

The Core Framework

Expression vs. Equation

The 5-Step Method

Example 1: Movie Tickets

Visualizing the Movie Ticket Problem

Example 2: Saving Money

Example 3: Two Unknowns, One Equation

The Key Insight in Example 3

Common Mistakes to Avoid

Summary: Problem Types at a Glance

Question 1 of 5

Question 2 of 5

Question 3 of 5

Question 4 of 5

Question 5 of 5

Linear Equation
Word Problems