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If $y$ varies directly with $x$ and $y = 16$ when $x = 2$ , 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 = 16

when

x = 2

, find

y

when

x = 1

. 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 $y = 16$ when $x = 2$ . Plugging these values into the direct variation equation gives us $16 = k \times 2$ .
Solve for k: Solve for k. $\newline$ Divide both sides of the equation by $2$ to isolate k. $\newline$ $\frac{16}{2} = \frac{k \times 2}{2}$ $\newline$ $k = 8$
Write equation with $k$ : Write the direct variation equation with the found value of $k$ . Now that we know $k = 8$ , the direct variation equation is $y = 8x$ .
Find $y$ for $x=1$ : Find $y$ when $x = 1$ . $\newline$ Substitute $x = 1$ into the direct variation equation $y = 8x$ . $\newline$ $y = 8 \times 1$ $\newline$ $y = 8$

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