Question

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

$-2x=y$

Short answer

The equation represents a direct variation.

Step-by-step solution

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

The relationship of the form $y=kx,x=\frac{y}{k},y=\frac{k}{x}$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. 

The relationship of the form $xy=k$is known as inverse variation. Observe that the given equation can be further simplified as:

$$\begin{gathered} -2x=y \\ \frac{-2x}{-2}=\frac{y}{-2} \\ x=-\frac{y}{2} \end{gathered}$$

is not comparable with$xy=k$ i.e., the product is not constant so the equation doesn’t represent the inverse variation.

Step 3. Check for direct variation.

The relationship of the form $y=kx$ is known as direct variation. Comparing the given equation $x=-\frac{y}{2}$ & $x=\frac{y}{k}$, observe that $k=-2$, therefore, the product is constant so the equation represents the direct variation.