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If $y$ varies directly with $x$ and $y = 12$ when $x = 4$ , find $y$ when $x = 1$ . 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 = 1

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

\newline

y

= ____

Establish Equation: Establish the direct variation equation. $\newline$ Since $y$ varies directly with $x$ , the equation can be represented as $y = kx$ , where $k$ is the constant of proportionality.
Find Constant $k$ : Use the given values to find the constant $k$ . We are given 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 Equation 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 $y$ for $x=1$ : Find $y$ when $x = 1$ using the direct variation equation. $\newline$ Substitute $x = 1$ into the equation $y = 3x$ to find $y$ . $\newline$ $y = 3 \times 1$ $\newline$ $y = 3$

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