Q3. - Algebra 1

Question

If $y$ varies inversely as $x$ and $y=4$ when $x=-8$ find $y$ when $x=-4$.

Step-by-Step Solution

Answer

When $x=-4,y=8$.

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 $x=-4$, substitute $-8$ for $x_1$, 4 for $y_1$, and $-4$ for $x_2$ into the equation $x_1{y}_1=x_2{y}_2$.

$$\begin{gathered} x_1{y}_1=x_2{y}_2 \\ \left(-8\right)\times 4=\left(-4\right)y_2 \end{gathered}$$

Step 3. Simplify for y 2 .

Further, simplify by dividing both sides by $-4$.

 $$\begin{gathered} (-8)\left(4\right)=(-4)y_2 \\ -32=-4y_2 \\ y_2=\frac{-32}{-4} \\ y_2=8 \end{gathered}$$

Therefore, $y=8$, when $x=-4$.