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An inverse variation includes the points $(4,\,4)$ and $(1,\,n)$ . 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

(4,\,4)

and

(1,\,n)

. Find

n

.

\newline

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

\newline

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

Identify general form of inverse variation: Given that there is an inverse variation that includes the points $(4, 4)$ and $(1, n)$ . Identify the general form of inverse variation. In inverse variation, the product of the two variables is constant. Inverse variation: $y = \frac{k}{x}$ where $k$ is the constant of variation.
Substitute values to find constant: We know that the point $(4, 4)$ lies on the inverse variation curve. $\newline$ Choose the equation after substituting the values in $y = \frac{k}{x}$ . $\newline$ Substitute $4$ for $x$ and $4$ for $y$ in $y = \frac{k}{x}$ to find the constant $k$ . $\newline$ $4 = \frac{k}{4}$
Solve for constant: 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 value of $n$ : Inverse variation equation: $y = \frac{16}{x}$ Find $y$ when $x = 1$ , which will give us the value of $n$ . Substitute $1$ for $x$ in $y = \frac{16}{x}$ . $n = \frac{16}{1}$ $n = 16$

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