Question

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

$x$	1	2	5
$y$	10	5	2

Short answer

The table represents an inverse 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,

$$\begin{gathered} 1\cdot 10=10 \\ 2\cdot 5=10 \\ 5\cdot 2=10 \end{gathered}$$

The product is constant so the table represents an inverse variation, $xy=10$.

Step 3. Check for direct variation.

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

$$\begin{gathered} \frac{10}{1}=10 \\ \frac{5}{2}=2.5 \\ \frac{2}{5}=0.4 \end{gathered}$$

Since$\frac{y}{x}$ the is not constant so the table doesn’t represent the direct variation.