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If $y$ varies directly with $x$ and $y = 14$ 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 = 14

when

x = 2

, find

y

when

x = 1

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

\newline

y

= ____

Select equation for direct variation: Select the equation that represents direct variation. $\newline$ Since $y$ varies directly with $x$ , we can use the direct variation equation: $y = kx$ .
Substitute values into equation: We know that $y = 14$ when $x = 2$ . Substitute the values into the direct variation equation $y = kx$ . Plug in $2$ for $x$ and $14$ for $y$ in $y = kx$ to find the constant of proportionality $k$ . $14 = k \times 2$
Solve for constant of proportionality: Solve for the constant of proportionality $k$ . $\newline$ Divide both sides by $2$ to isolate $k$ . $\newline$ $\frac{14}{2} = \frac{(k \times 2)}{2}$ $\newline$ $7 = k$
Write direct variation equation: Write the direct variation equation using the value of $k$ . $\newline$ Since we found that $k = 7$ , the direct variation equation is $y = 7x$ .
Find $y$ when $x = 1$ : Use the direct variation equation to find $y$ when $x = 1$ . Substitute $1$ for $x$ in the equation $y = 7x$ . $y = 7 \times 1$ $y = 7$

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