Variables Examples: 45 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Variables.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

A symbol (usually a letter like $x$ ) that represents an unknown or changing quantity in a mathematical expression.

Like a box that can hold any number. ' $x + 5 = 12$ ' asks: what's in the box?

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: A variable is a symbol that holds a number you don't know yet or one that can change.

Common stuck point: The procedure for variables is the easy part; the trap is treating $x$ as always meaning the same number across different problems. Asking "Is a letter being used to hold a number we don't know yet or one that can vary?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is a letter being used to hold a number we don't know yet or one that can vary?

Common Mistakes to Watch For

Before you work through the examples, skim the mistake guide so you know which shortcuts and sign errors to avoid.

Solving Linear Equations Mistakes

Worked Examples

Example 1

easy

x + 5 = 12

, what value does

x

represent?

Answer

x = 7

First step

The variable

x

is a placeholder for an unknown number.

Full solution

2
Subtract 5 from both sides: $x = 12 - 5 = 7$ .
3
Check: $7 + 5 = 12$ ✓

A variable is a symbol that stands for an unknown value. To find its value, we perform operations that isolate it on one side of the equation.

Example 2

medium

Evaluate the expression

3x + 2y

when

x = 4

and

y = -1

Example 3

medium

A rectangle has length

\ell

and width

w

. Write a formula for its perimeter

P

Example 4

medium

Maria is

3

years older than her brother. If her brother is

b

years old, how old is Maria?

Example 5

medium

x = 2y

and

y = 5

, what is

x

Example 6

hard

The sum of three consecutive integers is

48

. Let the smallest be

n

. Write and solve an equation for

n

Example 7

hard

5x - 3 = 2x + 9

, find

x

Example 8

challenge

Two numbers differ by

7

and their sum is

25

. Let the smaller be

x

. Find both numbers.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

2n = 18

, what is the value of

n

Example 2

medium

Write an expression using a variable for: 'a number increased by 8.'

Example 3

easy

In the expression

3n + 5

, what is the variable?

Example 4

easy

x = 4

, evaluate

x + 7

Example 5

easy

Write 'a number increased by

6

' using a variable.

Example 6

easy

2x

and

x \cdot 2

mean the same thing?

Example 7

easy

5y - 2

, what is the coefficient of

y

Example 8

easy

Let

n

be any integer. Is

n + 1

always the next integer?

Example 9

easy

Simplify

x + x + x

Example 10

easy

a

stands for apples and

b

for bananas, what does

a + b

count?

Example 11

medium

Marcus is

3

years older than Tina. If Tina is

t

years old, write Marcus's age.

Example 12

medium

A variable

p

represents a price in dollars. Why can't

p = -4

here?

Example 13

medium

x

doubles, what happens to

5x

Example 14

medium

Two unknowns satisfy

x + y = 10

. Is

x

a fixed number?

Example 15

medium

Express 'the sum of three consecutive integers' starting at

n

Example 16

medium

T = 2\pi\sqrt{L/g}

, which symbols are variables and which is a constant?

Example 17

medium

Why does

x - x = 0

for every value of

x

Example 18

medium

A recipe uses

c

cups of flour per loaf. Write the flour for

5

loaves.

Example 19

challenge

n

is even, explain why

n^2

is even using a variable.

Example 20

challenge

Distinguish the roles of

x

f(x) = x^2

versus

f(3) = 9

Example 21

challenge

A locker problem labels lockers

1

n

. Why use

n

instead of a number?

Example 22

medium

y = 3x

and

x

increases from

2

3

, how much does

y

increase?

Example 23

easy

y - 4 = 10

, what is

y

Example 24

easy

Evaluate

4x - 1

when

x = 3

Example 25

easy

Write an expression for 'a number

n

doubled, then added to

5

Example 26

medium

Evaluate

a^2 + b

when

a = -3

and

b = 5

Example 27

medium

p

is the price of one pencil in dollars, write an expression for the cost of

12

pencils.

Example 28

medium

Simplify:

5x - 2x + 4 - 1

Example 29

medium

A taxi charges a $3 flat fee plus $2 per mile. Write an expression for the cost of

m

miles.

Example 30

medium

Evaluate

\frac{2x + 6}{x}

when

x = 2

Example 31

hard

3x + 7 = 22

, find

x

Example 32

hard

Evaluate

x^2 - 2xy + y^2

when

x = 4, y = 1

Example 33

hard

A bag has

n

red marbles and

2n + 3

blue marbles. Write an expression for the total marbles.

Example 34

medium

Write an expression for 'the quotient of a number

n

and

4

, decreased by

1

Example 35

easy

Solve for

x

x + 11 = 20

Example 36

medium

Evaluate

\frac{x + y}{x - y}

when

x = 7, y = 3

Example 37

hard

A = \pi r^2

and

r = 3

, find

A

in terms of

\pi

← Back to Variables Formula Explained Common Mistakes Practice Mode

Related Concepts

Equal Number Sense Expressions Equations

Background Knowledge

These ideas may be useful before you work through the harder examples.

equalnumber sense