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If $y$ varies directly with $x$ and $y = 8$ when $x = 4$ , 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 = 8

when

x = 4

, find

y

when

x = 3

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

\newline

y

= ___

Establish equation for direct variation: Establish the equation for direct variation. $\newline$ Since $y$ varies directly with $x$ , the equation can be written as $y = kx$ , where $k$ is the constant of proportionality.
Find constant of proportionality: Use the given values to find the constant of proportionality $k$ . We are given that $y = 8$ when $x = 4$ . Plugging these values into the direct variation equation gives us $8 = k \times 4$ .
Solve for k: Solve for k. $\newline$ Divide both sides of the equation by $4$ to isolate k. $\newline$ $\frac{8}{4} = \frac{k \times 4}{4}$ $\newline$ $2 = k$
Write equation with $k$ : Write the direct variation equation with the found value of $k$ . Now that we know $k = 2$ , the direct variation equation is $y = 2x$ .
Find $y$ for $x=3$ : Find $y$ when $x = 3$ using the direct variation equation. $\newline$ Substitute $x = 3$ into the equation $y = 2x$ to find $y$ . $\newline$ $y = 2 \times 3$ $\newline$ $y = 6$

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