Variable as Placeholder Examples in Math

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\_ + 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=15x + 8 = 15, what specific value does xx represent?

Answer

x=7x = 7

First step

1
Here xx is a placeholder for one specific unknown number.

Full solution

  1. 2
    Subtract 8 from both sides: x=15−8=7x = 15 - 8 = 7.
  2. 3
    The placeholder xx 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 pp cents. Write an expression for the cost of 55 pencils, then find the cost when p=25p = 25.

Example 5

medium
Liam is three years older than his sister. Together their ages sum to 2525. Write an equation using a variable for the sister's age and solve.

Example 6

medium
A rectangle has perimeter 3030 cm and length 99 cm. Use a variable to find the width.

Example 7

medium
Maya buys xx notebooks at $3 each and 22 pens at $1.50 each. The total is $15. Find xx.

Example 8

hard
The sum of three consecutive integers is 4848. Use a variable to find them.

Example 9

hard
A father's age is 33 times his son's age. In 1010 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 44 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 99. 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 nn in 3n=213n = 21.

Example 2

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

Example 3

easy
In x+4=9x + 4 = 9, what does xx stand for?

Example 4

easy
Can the placeholder in 2n=72n = 7 be a fraction?

Example 5

easy
Does the letter have to be xx for a placeholder?

Example 6

easy
Fill the placeholder: □+3=8\square + 3 = 8. What number?

Example 7

easy
In A=lwA = lw, is ll a placeholder or a fixed number?

Example 8

easy
Substitute 55 for the placeholder nn in n2n^2.

Example 9

easy
How many values satisfy the placeholder in x+1=x+1x + 1 = x + 1?

Example 10

easy
In f(x)=x+2f(x) = x + 2, what role does xx play?

Example 11

medium
If □×4=△\square \times 4 = \triangle and □=3\square = 3, find △\triangle.

Example 12

medium
In the pattern 2,4,6,…,2n2, 4, 6, \dots, 2n, what does nn hold?

Example 13

medium
Solve for the placeholder: 3(□−2)=93(\square - 2) = 9.

Example 14

medium
Does xx in x>3x > 3 act as a single-value or range placeholder?

Example 15

medium
Two boxes: □+◯=10\square + \bigcirc = 10. Is □\square a single value?

Example 16

medium
In P=2(l+w)P = 2(l + w), solving for ll treats which letters as fixed?

Example 17

medium
A spreadsheet cell named AA holds a changing total. Placeholder or constant?

Example 18

challenge
In ∑i=1ni\sum_{i=1}^{n} i, what are the roles of ii and nn?

Example 19

challenge
Why is xx in ∫x dx\int x\,dx a 'dummy' placeholder?

Example 20

challenge
In a proof 'let nn be arbitrary', what placeholder role does nn play?

Example 21

medium
In T(n)=2n+1T(n) = 2n + 1, what does the placeholder nn receive?

Example 22

medium
Solve for the placeholder: □2+1=4\frac{\square}{2} + 1 = 4.

Example 23

easy
Find xx in x−6=10x - 6 = 10.

Example 24

easy
What number does the placeholder □\square represent in □⋅4=24\square \cdot 4 = 24?

Example 25

easy
In n+n+n=18n + n + n = 18, what is nn?

Example 26

easy
Substitute x=4x = 4 into 2x+32x + 3.

Example 27

medium
Solve 2x+7=212x + 7 = 21.

Example 28

medium
Solve x3−4=2\dfrac{x}{3} - 4 = 2.

Example 29

medium
Solve 3(x−2)=123(x - 2) = 12.

Example 30

medium
Solve 5x−2=3x+85x - 2 = 3x + 8.

Example 31

medium
A taxi charges a $4 starting fee plus $2 per mile. Let mm be the number of miles. Write an expression for the total cost and find mm when the total is $20.

Example 32

hard
Solve 2x−35=x+43\dfrac{2x - 3}{5} = \dfrac{x + 4}{3}.

Example 33

hard
Solve 4(x−1)−3(2x+1)=−114(x - 1) - 3(2x + 1) = -11.
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Background Knowledge

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

variables