Chapter 5, Problem 17CR

To determine

To find: whether $y$ directly with $x$ and write an equation for the direct variation.

Answer to Problem 17CR

The direct variation equation is $y=-2.5x$ .

Explanation of Solution

Given:

Concept used:

Directly proportional:

As one amount increases, another amount increases at the same rate.

Mathematically,

  $\begin{array}{l}y\alpha x.\\ y=kx.\\ \frac{y}{x}=k.\end{array}$

Where $k$ is the constant of proportionality.

Calculation:

Construct the following table:

$x$	$y$
$-3$	$7.5$
$-1$	$2.5$
$2$	$-5$
$5$	$-12.5$

Need to verify whether $y$ varies directly with $x$ .

A direct variation is a relationship that can be represented by a function in the form $y=kx$ where $k\ne 0$ .

So, the ratio of the variables is constant:

  $\frac{y}{x}=k$ .

Find the $\frac{y}{x}$ for each ordered pair using tabular values.

  $\begin{array}{l}\frac{7.5}{-3}=-\mathrm{2.5.}\\ \frac{2.5}{-1}=-\mathrm{2.5.}\\ \frac{-5}{2}=-\mathrm{2.5.}\\ \frac{-12.5}{5}=-\mathrm{2.5.}\end{array}$

The ratio $\frac{y}{x}=-2.5$ for each data pair.

So, $y$ varies directly with $x$ .

Hence, the direct variation equation is $y=-2.5x$ .