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Assume that $y$ varies inversely with $x$ . If $y = 2$ 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. Assume that

y

varies inversely with

x

. If

y = 2

when

x = 4

, find

y

when

x = 1

.

\newline

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

\newline

y =

_____

Understand Relationship: Understand the relationship between $y$ and $x$ . Since $y$ varies inversely with $x$ , the relationship can be described by the equation $y = \frac{k}{x}$ , where $k$ is the constant of variation.
Find Constant of Variation: Use the given values to find the constant of variation $k$ . We are given that $y = 2$ when $x = 4$ . Substitute these values into the inverse variation equation to find $k$ . $2 = \frac{k}{4}$
Solve for k: Solve for k. $\newline$ To find k, multiply both sides of the equation by $4$ . $\newline$ $2 \times 4 = k$ $\newline$ $8 = k$ $\newline$ Now we have found the constant of variation $k$ .
Write Inverse Variation Equation: Write the inverse variation equation with the found value of $k$ . Substitute $k = 8$ into the equation $y = \frac{k}{x}$ . $y = \frac{8}{x}$ This is the equation that describes the relationship between $y$ and $x$ .
Find $y$ for $x=1$ : Find $y$ when $x = 1$ . Substitute $x = 1$ into the inverse variation equation $y = \frac{8}{x}$ . $y = \frac{8}{1}$ $y = 8$

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