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A direct variation includes the points $(2,12)$ and $(1,n)$ . Find $n$ . Write and solve a direct variation equation to find the answer. $\newline$ $n = \_\_\_$

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Write and solve direct variation equations

Full solution

Q. A direct variation includes the points

(2,12)

and

(1,n)

. Find

n

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

\newline

n = \_\_\_

Identify direct variation: Which equation represents the direct variation? $\newline$ Since $y$ varies directly with $x$ , the direct variation can be represented by the equation $y = kx$ , where $k$ is the constant of variation.
Find constant of variation: Use the given point $(2,12)$ to find the constant of variation $k$ . Substitute $x = 2$ and $y = 12$ into the direct variation equation $y = kx$ to find $k$ . $12 = k \times 2$
Solve for k: Solve for k. $\newline$ Divide both sides by $2$ to isolate k. $\newline$ $12 / 2 = k$ $\newline$ $k = 6$
Write direct variation equation: Write the direct variation equation using the found value of $k$ . $\newline$ Since $k = 6$ , the direct variation equation is $y = 6x$ .
Find $n$ for $x=1$ : Use the direct variation equation to find $n$ when $x = 1$ . Substitute $x = 1$ into the equation $y = 6x$ to find $n$ . $n = 6 \times 1$ $n = 6$

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