Variable as Placeholder Examples: 44 Problems with Answers

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

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 variable acts as a placeholder — a letter or symbol (like x, n, or y) that stands in for an unknown or changing number in a mathematical expression or equation. It represents one specific unknown value that satisfies a given condition.

Like a blank in a sentence: ' $\_ + 3 = 7$ ' asks 'what number fits here?'

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-as-placeholder stands for one specific unknown value that a condition pins down.

Common stuck point: The procedure for variable as placeholder is the easy part; the trap is thinking the placeholder can be any number. Asking "Does the condition pin the variable to one specific value I'm meant to find?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Does the condition pin the variable to one specific value I'm meant to find?

Worked Examples

Example 1

easy

In the equation

x + 8 = 15

, what specific value does

x

represent?

Answer

x = 7

First step

Here

x

is a placeholder for one specific unknown number.

Full solution

2
Subtract 8 from both sides: $x = 15 - 8 = 7$ .
3
The placeholder $x$ represents exactly the value 7.

When a variable acts as a placeholder, it stands for one specific unknown value that we need to find. There is exactly one number that makes the equation true.

Example 2

medium

A rectangle has area 24 cm² and width 4 cm. Use a variable to find the length.

Example 3

medium

Maria has some stickers. After giving away 8, she has 15 left. Write an equation and find how many she started with.

Example 4

medium

A pencil costs

p

cents. Write an expression for the cost of

5

pencils, then find the cost when

p = 25

Example 5

medium

Liam is three years older than his sister. Together their ages sum to

25

. Write an equation using a variable for the sister's age and solve.

Example 6

medium

A rectangle has perimeter

30

cm and length

9

cm. Use a variable to find the width.

Example 7

medium

Maya buys

x

notebooks at $3 each and

2

pens at $1.50 each. The total is $15. Find

x

Example 8

hard

The sum of three consecutive integers is

48

. Use a variable to find them.

Example 9

hard

A father's age is

3

times his son's age. In

10

years, the father will be twice as old. Use a variable to find the son's current age.

Example 10

hard

Aiden has twice as many marbles as Ben. After Aiden gives

4

marbles to Ben, they have the same number. How many marbles did each have originally?

Example 11

challenge

A two-digit number's tens digit is twice its units digit, and the sum of the digits is

9

. Find the number.

Practice Problems

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

Example 1

easy

Find the value of

n

3n = 21

Example 2

medium

I think of a number, double it, and add 5 to get 19. What is my number?

Example 3

easy

x + 4 = 9

, what does

x

stand for?

Example 4

easy

Can the placeholder in

2n = 7

be a fraction?

Example 5

easy

Does the letter have to be

x

for a placeholder?

Example 6

easy

Fill the placeholder:

\square + 3 = 8

. What number?

Example 7

easy

A = lw

, is

l

a placeholder or a fixed number?

Example 8

easy

Substitute

5

for the placeholder

n

n^2

Example 9

easy

How many values satisfy the placeholder in

x + 1 = x + 1

Example 10

easy

f(x) = x + 2

, what role does

x

play?

Example 11

medium

\square \times 4 = \triangle

and

\square = 3

, find

\triangle

Example 12

medium

In the pattern

2, 4, 6, \dots, 2n

, what does

n

hold?

Example 13

medium

Solve for the placeholder:

3(\square - 2) = 9

Example 14

medium

Does

x

x > 3

act as a single-value or range placeholder?

Example 15

medium

Two boxes:

\square + \bigcirc = 10

. Is

\square

a single value?

Example 16

medium

P = 2(l + w)

, solving for

l

treats which letters as fixed?

Example 17

medium

A spreadsheet cell named

A

holds a changing total. Placeholder or constant?

Example 18

challenge

\sum_{i=1}^{n} i

, what are the roles of

i

and

n

Example 19

challenge

Why is

x

\int x\,dx

a 'dummy' placeholder?

Example 20

challenge

In a proof 'let

n

be arbitrary', what placeholder role does

n

play?

Example 21

medium

T(n) = 2n + 1

, what does the placeholder

n

receive?

Example 22

medium

Solve for the placeholder:

\frac{\square}{2} + 1 = 4

Example 23

easy

Find

x

x - 6 = 10

Example 24

easy

What number does the placeholder

\square

represent in

\square \cdot 4 = 24

Example 25

easy

n + n + n = 18

, what is

n

Example 26

easy

Substitute

x = 4

into

2x + 3

Example 27

medium

Solve

2x + 7 = 21

Example 28

medium

Solve

\dfrac{x}{3} - 4 = 2

Example 29

medium

Solve

3(x - 2) = 12

Example 30

medium

Solve

5x - 2 = 3x + 8

Example 31

medium

A taxi charges a $4 starting fee plus $2 per mile. Let

m

be the number of miles. Write an expression for the total cost and find

m

when the total is $20.

Example 32

hard

Solve

\dfrac{2x - 3}{5} = \dfrac{x + 4}{3}

Example 33

hard

Solve

4(x - 1) - 3(2x + 1) = -11

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Related Concepts

Variables Solving Linear Equations Variable as Generalization

Background Knowledge

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

variables