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Assume that $y$ varies inversely with $x$ . If $y = 4$ when $x = 8$ , find $y$ when $x = 2$ . $\newline$ Write and solve an inverse variation equation to find the answer. $\newline$ $y =$ _____

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Write and solve inverse variation equations

Full solution

Q. Assume that

y

varies inversely with

x

. If

y = 4

when

x = 8

, find

y

when

x = 2

.

\newline

Write and solve an inverse variation equation to find the answer.

\newline

y =

_____

Identify general form: Given that $y$ varies inversely with $x$ . Identify the general form of inverse variation. In inverse variation, variables change in opposite directions. Inverse variation: $y = \frac{k}{x}$
Substitute values: We know that $y = 4$ when $x = 8$ . Choose the equation after substituting the values in $y = \frac{k}{x}$ . Substitute $8$ for $x$ and $4$ for $y$ in $y = \frac{k}{x}$ . $4 = \frac{k}{8}$
Solve for k: We found: $\newline$ $4 = \frac{k}{8}$ $\newline$ Solve the equation to find the value of k. $\newline$ To isolate k, multiply both sides by $8$ . $\newline$ $4 \times 8 = \left(\frac{k}{8}\right) \times 8$ $\newline$ $32 = k$
Write inverse variation: We have: $\newline$ $k = 32$ $\newline$ Write the inverse variation equation in the form of $y = \frac{k}{x}$ . $\newline$ Substitute $k = 32$ in $y = \frac{k}{x}$ . $\newline$ $y = \frac{32}{x}$
Find $y$ for $x=2$ : Inverse variation equation: $\newline$ $y = \frac{32}{x}$ $\newline$ Find $y$ when $x = 2$ . $\newline$ Substitute $2$ for $x$ in $y = \frac{32}{x}$ . $\newline$ $y = \frac{32}{2}$ $\newline$ $y = 16$

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