Algebraic Representation Examples in Math

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

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

Concept Recap

Using algebraic expressions and equations to represent and analyze mathematical relationships and real-world situations.

Translating 'the cost is 5 plus 2 per item' into C = 5 + 2n.

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: Algebra provides a precise language for describing patterns and relationships.

Common stuck point: Translating words to algebraic symbols requires careful reading — identify quantities, operations, and whether an expression or equation is needed.

Sense of Study hint: Underline the key words in the problem and write the matching math symbol above each one.

Worked Examples

Example 1

easy
Represent this situation algebraically: 'Sarah has 5 more apples than Tom. Together they have 23 apples.'

Solution

  1. 1
    Let t = Tom's apples. Then Sarah has t + 5 apples.
  2. 2
    Together: t + (t + 5) = 23.
  3. 3
    Simplify: 2t + 5 = 23, so 2t = 18 and t = 9.
  4. 4
    Tom has 9 apples; Sarah has 14 apples.

Answer

t + (t+5) = 23; Tom: 9, Sarah: 14.
Algebraic representation translates words into symbols. Choose a variable for an unknown quantity, express other quantities in terms of it, and write an equation from the given relationship.

Example 2

medium
Write an algebraic formula for the perimeter of a rectangle with length \ell and width w.

Practice Problems

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

Example 1

easy
Write an expression for the total cost of n books at \$8 each.

Example 2

medium
A taxi charges \3 base fare plus \2 per mile. Write a formula for the fare F in terms of miles m.
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Background Knowledge

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

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