Question

Question: Solve. Assume that y varies inversely as $x$.

If$y=7$when$x=6$finding $y$when$x=-21$.

Short answer

When $x=-21,y=-2$.

Step-by-step solution

Step 1. State the concept. 

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 $x$.

Product Rule for Inverse Variations – If $\left(x_1,y_1\right)$and $\left(x_2,y_2\right)$are solutions of an inverse variation, then $x_1{y}_1=x_2{y}_2$.

Step 2. Apply Product rule for Inverse Variations. 

To find y when $x=-21$, substitute 6 for $x_1$, 7 for $y_1$, and $-21$for $x_2$into the equation $x_1{y}_1=x_2{y}_2$.

$$\begin{gathered} x_1{y}_1=x_2{y}_2 \\ \left(6\right)\times 7=\left(-21\right)y_2 \end{gathered}$$

Step 3. Simplify for y2.

Further, simplify by dividing both sides by $-21$.

$\left(6\right)\times 7=\left(-21\right)y_2$

$$\begin{gathered} 42=-21y_2 \\ {y}_2=\frac{42}{-21} \\ {y}_2=-2 \end{gathered}$$

Therefore, $y=-2$, when $x=-21$.