Q10. - Algebra 1

Question

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

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

Step-by-Step Solution

Answer

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

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 y 2 .

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 $x=-21$.