Linear Relationship Examples in Math

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

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 relationship where quantities change at a constant rate, graphing as a straight line.

Add the same amount each step. Like paying \10$/month—increase is constant.

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: Linear means constant rate of change—the graph is a straight line.

Common stuck point: Linear relationships can have different starting points (y = mx + b).

Sense of Study hint: Calculate the difference between consecutive y-values; if it is always the same, the relationship is linear.

Worked Examples

Example 1

easy
A taxi charges a \3 base fee plus \2 per mile. Write the equation for total cost \(C\) in terms of miles \(m\). Identify slope and y-intercept.

Solution

  1. 1
    Base fee (y-intercept): \(b = 3\).
  2. 2
    Cost per mile (slope): \(m_{\text{rate}} = 2\).
  3. 3
    Equation: \(C = 2m + 3\).
  4. 4
    This is in the form \(y = mx + b\) with slope 2 and y-intercept 3.

Answer

\(C = 2m + 3\); slope = 2, y-intercept = 3
A linear relationship \(y = mx + b\) has constant slope \(m\) (rate of change) and y-intercept \(b\) (starting value).

Example 2

medium
Two points on a line are \((1, 5)\) and \((3, 11)\). Find the equation of the line in \(y = mx + b\) form.

Practice Problems

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

Example 1

easy
For the equation \(y = 4x - 1\), find \(y\) when \(x = 3\) and \(x = 0\).

Example 2

medium
Points \((0, -2)\) and \((4, 6)\) lie on a line. Find the equation in \(y = mx + b\) form.
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Background Knowledge

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

rate of change