Question

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

If$y=-6$ when $x=9$, find x when $y=-3$.

Short answer

The direct variation equation that relates x and y is$y=\frac{-2}{3}x$ and the value of x when$y=-3$ is $\frac{9}{2}$.

Step-by-step solution

Step 1. Write a direct variation equation that relates x and y.

It is given that y varies directly as x.

Therefore it implies that $y\alpha x$.

Therefore, it is obtained that:

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

Where k is constant of proportionality.

It is given that when $x=9,y=-6$.

Therefore, substitute 9 for x and $-6$ for y in the equation $y=kx$ to find the value of k.

$\begin{array}{c}y=kx\\ -6=k\left(9\right)\\ \frac{-6}{9}=k\\ \frac{-2}{3}=k\end{array}$

Substitute the value of k in the equation $y=kx$.

Therefore, it is obtained that:

$\begin{array}{c}y=kx\\ y=\frac{-2}{3}x\end{array}$

Therefore, the direct variation equation that relates x and y is $y=\frac{-2}{3}x$.

Step 2. Find the value of x when y=−3.

The direct variation equation that relates x and y is $y=\frac{-2}{3}x$.

Find the value of x by substituting $-3$ for y in the equation $y=\frac{-2}{3}x$.

$\begin{array}{c}y=\frac{-2}{3}x\\ -3=\frac{-2}{3}x\\ -9=-2x\\ \frac{-9}{-2}=x\\ \frac{9}{2}=x\end{array}$

Therefore, the value of x when$y=-3$ is $\frac{9}{2}$.