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 $y=12$.

Short answer

The direct variation equation that relates x and y is$y=\frac{2}{3}x$ and the value of x when$y=12$ is 18.

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:

$$\begin{gathered} y\alpha x \\ y=kx \end{gathered}$$

Where is k 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.

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

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=12.

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

Find the value of x by substituting 12 for y in the equation $y=\frac{2}{3}x$.

$$\begin{gathered} y=\frac{2}{3}x \\ 12=\frac{2}{3}x \\ 12\times 3=2x \\ 36=2x \\ \frac{36}{2}=x \\ 18=x \end{gathered}$$

Therefore, the value of x when$y=12$ is 18.