EQN-03-13

Direct & Inverse Variation

Model "y varies as x" relationships with a constant k, and use one data point to find the rest.

~5 min read•Difficulty 2/5

"y varies directly as x" means $y = k \cdot x$ for a constant k. Their ratio y/x is fixed.

Find k from one pair, then use it .

y = 12 when x = 3 \to k = 12/3 = 4 \to y = 4 x

"y varies inversely as x" means $y = k / x,$ so the product $x \cdot y$ is fixed.

y = 6 when x = 4 \to k = 6 \times 4 = 24 \to y = 24/ x

This is the algebraic twin of the EQN-02 proportion lesson — same idea, with an explicit k.

Worked examples

Example 1 (difficulty 2/5)

y varies directly as x, and

y = 20

when

x = 5.

Find y when

x = 8.

◎ Predict the answer first:

Example 2 (difficulty 2/5)

y varies inversely as x, and

y = 9

when

x = 2.

Find y when

x = 6.

◎ Predict the answer first:

Direct $: y =$ kx (ratio fixed). Inverse $: y = k / x ($ product fixed).
Always find k from the first data pair, then reuse it.
Inverse-variation answers move opposite to the input — sanity-check the direction.

Using

y =

kx for an inverse-variation problem

Why: Direct variation is the default model.

Fix: When one quantity falls as the other rises, use

y = k / x .

Forgetting to compute k before answering

Why: Rushing to the new value.

Fix: Step 1 is always finding k from the given pair.