Direct Variation Formula

The Formula

y = kx \quad (k \neq 0)

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

Quick Example

If y varies directly with x and y = 12 when x = 3, then y = 4x

Notation

'y varies directly as x' or 'y is directly proportional to x'

What This Formula Means

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.

Formal View

y \propto x \iff \exists\, k \neq 0: y = kx, \; \text{so } (0,0) \text{ is always a solution}

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.

Common Mistakes

  • Calling y = 3x + 1 a direct variation — direct variation requires b = 0 so the line passes through the origin
  • Confusing direct variation with any linear equation — all direct variations are linear, but not all linear equations are direct variations
  • Forgetting to check whether (0, 0) is a solution — if x = 0 gives y \neq 0, it is not direct variation

Why This Formula Matters

Direct variation is the simplest proportional relationship and graphs as a straight line through the origin. It models constant-speed motion, unit pricing, and currency conversion, making it one of the most common relationships in science and daily life.

Frequently Asked Questions

What is the Direct Variation formula?

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.

How do you use the Direct Variation formula?

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

What do the symbols mean in the Direct Variation formula?

'y varies directly as x' or 'y is directly proportional to x'

Why is the Direct Variation formula important in Math?

Direct variation is the simplest proportional relationship and graphs as a straight line through the origin. It models constant-speed motion, unit pricing, and currency conversion, making it one of the most common relationships in science and daily life.

What do students get wrong about Direct Variation?

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

What should I learn before the Direct Variation formula?

Before studying the Direct Variation formula, you should understand: proportionality, linear relationship.

← Back to Direct Variation See All Examples Practice Problems