Direct Variation Examples in Math

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

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 proportional relationship of the form y = kx (where k \neq 0) that always passes through the origin; when one quantity doubles, the other doubles, and the ratio \frac{y}{x} remains constant.

Distance varies directly with time at constant speed: d = 60t.

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: Direct variation goes through the origin—when x = 0, y = 0.

Common stuck point: y = 2x + 3 is NOT direct variation (doesn't pass through origin).

Sense of Study hint: Check whether the point (0, 0) fits the relationship; if it does not, it is not direct variation.

Worked Examples

Example 1

easy
\(y\) varies directly with \(x\), and \(y = 18\) when \(x = 3\). Find the constant \(k\) and write the direct variation equation.

Solution

  1. 1
    Direct variation: \(y = kx\).
  2. 2
    Find \(k\): \(k = y/x = 18/3 = 6\).
  3. 3
    Equation: \(y = 6x\).
  4. 4
    Check: when \(x=3\), \(y = 6 \times 3 = 18\) ✓

Answer

\(k = 6\); \(y = 6x\)
In direct variation \(y = kx\), \(k\) is found by dividing \(y\) by \(x\). Here \(k = 18/3 = 6\).

Example 2

medium
The cost of fabric varies directly with length. 5 meters costs \$35. How much do 8 meters cost? Set up a proportion.

Practice Problems

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

Example 1

easy
If \(y = kx\) and \(y = 24\) when \(x = 4\), find \(y\) when \(x = 7\).

Example 2

medium
A machine produces 150 units in 5 hours. Assuming direct variation, how many units in 9 hours?
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Background Knowledge

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

proportionalitylinear relationship