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If $y$ varies directly with $x$ and $y = 4$ 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 = 4

when

x = 2

, find

y

when

x = 1

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

\newline

y

= ____

Determine Equation: Determine the equation that represents the direct variation. $\newline$ Since $y$ varies directly with $x$ , the relationship can be described by the equation $y = kx$ , 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 $y = 4$ when $x = 2$ . Plugging these values into the direct variation equation gives us $4 = k \times 2$ .
Solve for k: Solve for k. $\newline$ Dividing both sides of the equation by $2$ gives us $k = \frac{4}{2}$ , which simplifies to $k = 2$ .
Write Equation with $k$ : Write the direct variation equation using the found value of $k$ . Now that we know $k = 2$ , the direct variation equation is $y = 2x$ .
Find $y$ for $x = 1$ : Find $y$ when $x = 1$ using the direct variation equation. $\newline$ Substitute $x = 1$ into the equation $y = 2x$ to get $y = 2 \times 1$ , which simplifies to $y = 2$ .

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