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An inverse variation includes the points $(8,\,2)$ and $(4,\,n)$ . Find $n$ . $\newline$ Write and solve an inverse variation equation to find the answer. $\newline$ $n = \,\_\_\_\_$

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Write and solve inverse variation equations

Full solution

Q. An inverse variation includes the points

(8,\,2)

and

(4,\,n)

. Find

n

.

\newline

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

\newline

n = \,\_\_\_\_

Identify general form: Given that there is an inverse variation between two variables. $\newline$ Identify the general form of inverse variation. $\newline$ In inverse variation, one variable is directly proportional to the reciprocal of the other. $\newline$ Inverse variation: $y = \frac{k}{x}$
Find constant of variation: We know that the point $(8, 2)$ lies on the inverse variation curve. $\newline$ Use this point to find the constant of variation $k$ . $\newline$ Substitute $8$ for $x$ and $2$ for $y$ in $y = k / x$ . $\newline$ $2 = k / 8$
Solve for k: Solve the equation to find the value of $k$ . To isolate $k$ , multiply both sides by $8$ . $2 \times 8 = \left(\frac{k}{8}\right) \times 8$ $16 = k$
Write inverse variation equation: We have found the constant of variation: $\newline$ $k = 16$ $\newline$ Write the inverse variation equation using the value of $k$ . $\newline$ $y = \frac{16}{x}$
Find $n$ : We need to find $n$ when $x = 4$ . $\newline$ Substitute $4$ for $x$ in $y = \frac{16}{x}$ to find $n$ . $\newline$ $n = \frac{16}{4}$ $\newline$ $n = 4$

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