Question

Determine whether each table or equation represents an inverse or a direct variation. Explain.

$x$	2	3	4	5
$y$	$-6$	$-9$	$-12$	$-15$

Short answer

The table represents a direct variation.

Step-by-step solution

Step 1. State the concept of direct variation and inverse variation.

The relationship of the form $y=kx$is called direct variation.

Relationship of the form $xy=k$or $y=\frac{k}{x}$ where $x,y\ne 0$ for some nonzero constant$k$ is known as inverse variation i.e., $y$varies inversely as $x$.

Step 2. Check for inverse variation. 

Find $xy$each ordered pair from the table and get,

role="math" localid="1648109306727" $\begin{array}{c}2\cdot (-6)=-12\\ 3\cdot (-9)=-27\\ 4\cdot (-12)=-48\\ 5\cdot (-15)=-75\end{array}$

The product is not constant so the table doesn’t represent an inverse variation.

Step 3. Check for direct variation.

Find $\frac{y}{x}$each ordered pair from the table and get,

$\begin{array}{c}\frac{-6}{2}=-3\\ \frac{-9}{3}=-3\\ \frac{-12}{4}=-3\\ \frac{-15}{5}=-3\end{array}$

Since, the $\frac{y}{x}=-3$, for all given values of $x$ and $y$, so the table represents the direct variation of $y=-3x$ .