Direct Variation

Finding $k$

Since $y = kx$, dividing both sides by $x$ gives $k = \dfrac{y}{x}$. One pair of values is all you need: divide $y$ by $x$ and you have the constant.

In a table, direct variation shows itself as a constant ratio. Divide each $y$ by its $x$ — if every row gives the same number, the table is a direct variation and that number is $k$. If the ratios differ, it is not direct variation, no matter how linear the table looks.

The table below is a direct variation: every $y \div x$ gives the same constant, here $4$.

Writing the equation and making predictions

Once you know $k$, write the equation $y = kx$ and it answers every question about the relationship. Given a new $x$, multiply by $k$ to predict $y$. Given a new $y$, divide by $k$ to recover $x$.

This two-step pattern — find $k$ from one pair, then reuse it on another — is the whole skill. Every direct variation word problem is those two steps wearing different clothes.

Direct variation vs. other lines

Every direct variation is linear, but not every line is a direct variation. The line $y = 3x + 2$ has a constant slope, but it crosses the $y$-axis at $2$, not at the origin — so $y$ is not a constant multiple of $x$. Only lines of the form $y = kx$, passing through $(0, 0)$, qualify.

Below, the solid line $y = 2x$ is a direct variation — it runs straight through the origin — while the dashed line $y = 2x + 2$ has the same slope but crosses the $y$-axis at $2$, so it is not.

Worked examples

Try one yourself

Common questions