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If $y$ varies directly with $x$ and $y = 12$ 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 = 12

when

x = 4

, find

y

when

x = 3

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

\newline

y

= ____

Write Equation: Write the equation that represents the direct variation between $y$ and $x$ . Since $y$ varies directly with $x$ , the equation can be written as $y = kx$ , where $k$ is the constant of variation.
Find Constant: Use the given values to find the constant of variation $k$ . We know that $y = 12$ when $x = 4$ . Substitute these values into the direct variation equation to find $k$ . $12 = k \times 4$
Solve for k: Solve for k. $\newline$ Divide both sides of the equation by $4$ to isolate k. $\newline$ $k = \frac{12}{4}$ $\newline$ $k = 3$
Write with $k$ : Write the direct variation equation with the found value of $k$ . Now that we know $k = 3$ , the direct variation equation is $y = 3x$ .
Find $y$ : Use the direct variation equation to find $y$ when $x = 3$ . Substitute $x = 3$ into the equation $y = 3x$ to find $y$ . $y = 3 \times 3$ $y = 9$

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