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An inverse variation includes the points $(8,\,1)$ 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

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Q. An inverse variation includes the points

(8,\,1)

and

(4,\,n)

. Find

n

.

\newline

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

\newline

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

Understand Inverse Variation: Understand the concept of inverse variation. $\newline$ In an inverse variation, the product of the two variables is constant. This means that if one variable increases, the other decreases in such a way that their product remains the same.
Write Equation Using Points: Use the given points to write the inverse variation equation. $\newline$ The general form of an inverse variation is $y = \frac{k}{x}$ , where $k$ is the constant of variation. We are given the points $(8, 1)$ and $(4, n)$ , which means when $x = 8$ , $y = 1$ , and when $x = 4$ , $y = n$ .
Find Constant of Variation: Find the constant of variation using the first point $(8, 1)$ . $\newline$ Substitute $x = 8$ and $y = 1$ into the inverse variation equation $y = \frac{k}{x}$ . $\newline$ $1 = \frac{k}{8}$ $\newline$ Now, solve for $k$ by multiplying both sides by $8$ . $\newline$ $1 \times 8 = k$ $\newline$ $k = 8$
Use Constant to Find $n$ : Use the constant of variation to find $n$ using the second point $(4, n)$ . $\newline$ Now that we know $k = 8$ , we can use the second point $(4, n)$ to find $n$ . Substitute $k = 8$ and $x = 4$ into the inverse variation equation $y = \frac{k}{x}$ . $\newline$ $n = \frac{8}{4}$ $\newline$ Now, solve for $n$ . $\newline$ $n$ $1$

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