Modeling with Equations Examples in Math

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

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

Concept Recap

Translating a real-world situation into one or more equations that capture its mathematical relationships and constraints.

Turn a word problem into math: identify what's unknown, write relationships as equations.

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: Equations model constraints; solving finds values that satisfy them.

Common stuck point: Clearly defining what the variable represents (with units) before writing any equation is the most skipped step.

Sense of Study hint: Write down 'Let x = ...' as your first sentence, clearly defining what the variable represents in context.

Worked Examples

Example 1

easy
A phone plan costs \15 per month plus \0.10 per text. Write an equation for the monthly cost C if you send t texts.

Solution

  1. 1
    Fixed cost: \$15 per month.
  2. 2
    Variable cost: \0.10 per text, so 0.10t for t$ texts.
  3. 3
    Total: C = 15 + 0.10t.

Answer

C = 15 + 0.10t
Modeling translates a real situation into an equation. Identify the fixed part (constant) and the changing part (variable term) to build the equation.

Example 2

medium
Two trains leave the same station traveling in opposite directions at 50 mph and 70 mph. After how many hours are they 360 miles apart?

Practice Problems

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

Example 1

easy
Write an equation: 'A number doubled and increased by 7 equals 25.'

Example 2

hard
The sum of three consecutive integers is 72. Find them.
← Back to Modeling with Equations Formula Explained Practice Mode

Background Knowledge

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

equationsalgebraic representation