Direct variation

The idea

y varies directly with x when

y = kx

where k is the constant of variation (k ≠ 0). If x doubles, y doubles; if x is cut in half, y is cut in half.

Features

On the graph, k is the slope: the rise for every 1 unit of run.

Finding k

Divide y by x for any known pair.

Example: y varies directly with x, and y = 24 when x = 6.

• k = 24 ÷ 6 = 4
• Equation: y = 4x
• When x = 10: y = 40

Real-life direct variation

Using proportions

Because y₁/x₁ = y₂/x₂, you can solve direct variation problems with a proportion.

Example: A spring stretches 6 cm under a 4 kg load. How far under 10 kg?

• 6/4 = s/10
• s = 15 cm

Direct vs partial variation

A plumber charging a fixed fee plus an hourly rate is not direct variation.

Worked examples

Example 1: y varies directly with x; y = 15 when x = 5. Find y when x = 12.

k = 3 → y = 36.

Example 2: Is y = 2x + 1 a direct variation?

No; it doesn’t pass through (0, 0).

Example 3: Cost varies directly with mass. 3 kg cost $7.50. Cost of 8 kg?

k = 2.50 → $20.00.

Common mistakes

• Treating any straight line as direct variation.
• Finding k as x/y instead of y/x.
• Forgetting units in k (e.g. $/kg).

Try it yourself

1. y = 18 when x = 3. Write the equation.
2. Using it, find y when x = 7.
3. Is the table x: 2, 4, 6 and y: 7, 14, 21 a direct variation?
4. A car uses 9 L for 120 km. How much for 300 km?

Answers: 1) y = 6x 2) 42 3) yes, k = 3.5 4) 22.5 L

Quick recap

• Direct variation: y = kx; the graph passes through the origin.
• k = y/x is the constant of variation and the slope.
• If there’s a starting amount, it’s partial variation.