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If $y$ varies inversely with $x$ and $y = 4$ when $x = 4$ , find $y$ when $x = 1$ . $\newline$ 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. If

y

varies inversely with

x

and

y = 4

when

x = 4

, find

y

when

x = 1

.

\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 in equation: We know that $y = 4$ when $x = 4$ . Choose the equation after substituting the values in $y = \frac{k}{x}$ . Substitute $4$ for $x$ and $4$ for $y$ in $y = \frac{k}{x}$ . $4 = \frac{k}{4}$
Solve for $k$ : We found: $\newline$ $4 = \frac{k}{4}$ $\newline$ Solve the equation to find the value of $k$ . $\newline$ To isolate $k$ , multiply both sides by $4$ . $\newline$ $4 \times 4 = \left(\frac{k}{4}\right) \times 4$ $\newline$ $16 = k$
Write inverse variation equation: We have: $\newline$ $k = 16$ $\newline$ Write the inverse variation equation in the form of $y = \frac{k}{x}$ . $\newline$ Substitute $k = 16$ in $y = \frac{k}{x}$ . $\newline$ $y = \frac{16}{x}$
Find $y$ for $x=1$ : Inverse variation equation: $\newline$ $y = \frac{16}{x}$ $\newline$ Find $y$ when $x = 1$ . $\newline$ Substitute $1$ for $x$ in $y = \frac{16}{x}$ . $\newline$ $y = \frac{16}{1}$ $\newline$ $y = 16$

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