Q11. - Algebra 1

Question

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

If $y=-5$ when $x=9$ finding $y$ when $x=6$.

Step-by-Step Solution

Answer

When $x=6,y=-7.5$.

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 $x=6$, substitute 9 for $x_1$, $-5$ for $y_1$, and 6 for $x_2$ into the equation $x_1{y}_1=x_2{y}_2$.

$$\begin{gathered} x_1{y}_1=x_2{y}_2 \\ \left(9\right)\times \left(-5\right)=\left(6\right)y_2 \end{gathered}$$

Step 3. Simplify for y 2 .

Further, simplify by dividing both sides by 6.

$\left(9\right)\times \left(-5\right)=\left(6\right)y_2$

        $$\begin{gathered} -45=6y_2 \\ y_2=\frac{-45}{6} \\ y_2=-\text{7.5} \end{gathered}$$

Therefore, $y=-\text{7.5}$, when $x=6$.