Question

Suppose yvaries directly as x. Write a direct variation equation that relates x and y. Then solve

If$y=2$when$x=-12$, what is y when$x=-4$?

Short answer

The value of y when $x=-4$is$\frac{2}{3}.$

Step-by-step solution

Step 1. State the concept of Direct Proportionality.

Direct proportion is a mathematical comparison between two numbers where the ratio of the two numbers is equal to a constant value. The proportion definition says that when two ratios are equivalent, they are in proportion. The symbol used to relate the proportions is "$\propto$".

Here yvaries directly as x.

That is, $y\propto x$

Therefore$y=kx\dots \left(1\right)$

Where, kis a constant.

Step 2. Find the value of ‘k’.

Find the value of k by substituting$y=2\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}x=-12$ in equation$y=kx$

$\begin{array}{c}2=k\left(-12\right)\\ \frac{2}{-12}=\frac{-12k}{-12}\\ \frac{1}{-6}=k\\ k=\frac{1}{-6}\left(\text{Equation\hspace{0.17em}\hspace{0.17em}}2\right)\end{array}$

Step 3. Calculate the value of ‘y’

Calculate the value of y by substituting $x=-4\text{\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}}y=\frac{1}{-6}$in $y=kx$

$$\begin{gathered} y=\left(\frac{1}{-6}\right)\left(-4\right) \\ y=\frac{2}{3} \end{gathered}$$

Thus, the value of y is $\frac{2}{3}$.