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If $y$ varies directly with $x$ and $y = 20$ when $x = 5$ , find $y$ when $x = 3$ . Write and solve a direct variation equation to find the answer. $\newline$ $y$ = ____

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Write and solve direct variation equations

Full solution

Q. If

y

varies directly with

x

and

y = 20

when

x = 5

, find

y

when

x = 3

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

\newline

y

= ____

Establish direct variation equation: Establish the direct variation equation. $\newline$ Since $y$ varies directly with $x$ , the equation can be written as $y = kx$ , 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 $y = 20$ when $x = 5$ . Substituting these values into the direct variation equation gives us $20 = k \times 5$ .
Solve for k: Solve for k. $\newline$ Divide both sides of the equation by $5$ to isolate $k$ . $\newline$ $\frac{20}{5} = \frac{(k \times 5)}{5}$ $\newline$ $k = 4$
Write equation with $k$ : Write the direct variation equation with the found value of $k$ . Now that we know $k = 4$ , the direct variation equation is $y = 4x$ .
Find $y$ for $x=3$ : Find $y$ when $x = 3$ . $\newline$ Substitute $x = 3$ into the direct variation equation $y = 4x$ . $\newline$ $y = 4 \times 3$ $\newline$ $y = 12$

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