Q15. - Algebra 1

Question

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

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

Step-by-Step Solution

Answer

The table represents a direct variation.

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,

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

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 $y=-3x$ .