Direct and Inverse Proportion: Practice

The variable $x$ sits in the denominator, matching the form $y = \dfrac{k}{x}$ with $k = 18$.

$xy = 18$

The product is constant, which is inverse variation.

Substitute $x = 7$ into $y = 3x$.

$y = 3 \times 7 = 21$

Direct variation is $y = kx$. Replacing $x$ with $2x$ gives $k(2x) = 2(kx)$.

$y_{\text{new}} = 2kx = 2y$

So $y$ doubles.

Direct proportion means $y$ is a constant multiple of $x$.

$y = kx$

Adding or dividing would not keep the ratio $y/x$ constant.

Inverse proportion puts $x$ in the denominator so the product stays constant.

$y = \frac{k}{x} \quad\Rightarrow\quad xy = k$

A constant ratio $y/x$ is the signature of direct variation, and that constant is $k$.

$\frac{y}{x} = 5 \quad\Rightarrow\quad y = 5x$

So it is direct with $k = 5$.

Set $x = 0$ in $y = kx$.

$y = k \times 0 = 0$

So the line always passes through $(0, 0)$, the origin.

First find $k = \dfrac{y}{x}$ from the known pair.

$k = \frac{12}{3} = 4$

Then $y = 4x = 4 \times 5 = 20$.

Inverse variation means $xy = k$.

$k = 6 \times 4 = 24$

In inverse variation, one quantity rises as the other falls, with a constant product. More workers on a fixed job means less time, and (workers) times (time) is the fixed amount of work.

$(\text{workers}) \times (\text{time}) = \text{constant}$

The other three rise together, which is direct variation.

First find the constant $k = xy$ from the known pair.

$k = 2 \times 9 = 18$

Then $y = \dfrac{18}{x} = \dfrac{18}{3} = 6$.

A constant ratio $y/x$ means $y = kx$ for a fixed $k$.

$\frac{y}{x} = k$

That is the definition of direct variation.