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If $y$ varies directly with $x$ and $y = 20$ when $x = 10$ , find $y$ when $x = 4$ . Write and solve a direct variation equation to find the answer. $\newline$ $y = \underline{\hspace{2em}}$

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Write and solve direct variation equations

Full solution

Q. If

y

varies directly with

x

and

y = 20

when

x = 10

, find

y

when

x = 4

. Write and solve a direct variation equation to find the answer.

\newline

y = \underline{\hspace{2em}}

Select equation: Select the equation that represents direct variation. $\newline$ Since $y$ varies directly with $x$ , we can use the direct variation equation: $y = kx$ , where $k$ is the constant of proportionality.
Find $k$ : Use the given values to find the constant of proportionality ( $k$ ). $\newline$ We are given that $y = 20$ when $x = 10$ . Substitute these values into the direct variation equation to find $k$ . $\newline$ $20 = k \times 10$
Solve for k: Solve for k. $\newline$ Divide both sides of the equation by $10$ to isolate k. $\newline$ $\frac{20}{10} = \frac{k \times 10}{10}$ $\newline$ $k = 2$
Write equation: Write the direct variation equation using the value of $k$ . $\newline$ Now that we know $k = 2$ , we can write the direct variation equation as $y = 2x$ .
Find $y$ : Use the direct variation equation to find $y$ when $x = 4$ . Substitute $x = 4$ into the equation $y = 2x$ to find $y$ . $y = 2 \times 4$ $y = 8$

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