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If $x$ and $y$ are in direct proportion and $y$ is $36$ when $x$ is $9$ , find $y$ when $x$ is $8$ .

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Write and solve direct variation equations

Full solution

Q. If

x

and

y

are in direct proportion and

y

is

36

when

x

is

9

, find

y

when

x

is

8

.

Write Equation: Write the equation that represents direct variation. $\newline$ Since $y$ varies directly with $x$ , we can write the equation as $y = kx$ , where $k$ is the constant of proportionality.
Find Constant: Use the given values to find the constant of proportionality $k$ . We know that $y = 36$ when $x = 9$ . Substitute these values into the direct variation equation to find $k$ . $36 = k \times 9$
Solve for k: Solve for k. $\newline$ Divide both sides of the equation by $9$ to isolate k. $\newline$ $k = \frac{36}{9}$ $\newline$ $k = 4$
Use Value of $k$ : Write the direct variation equation using the value of $k$ . Now that we know $k = 4$ , the direct variation equation is $y = 4x$ .
Find $y$ : Find $y$ when $x = 8$ . $\newline$ Substitute $x = 8$ into the direct variation equation to find $y$ . $\newline$ $y = 4 \times 8$ $\newline$ $y = 32$

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