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The quantity $z$ varies directly with $y$ and inversely with $w$ and $x$ . When $y = 12$ , $w = 8$ , and $x = –5$ , $z = –3$ . What is the value of $k$ , the constant of variation? $\newline$ Write your answer in the form $A$ or $\frac {(A)}{(B)}$ , where $A$ and $B$ are constants or variable expressions. $\newline$ ____

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Find the constant of variation

Full solution

Q. The quantity

z

varies directly with

y

and inversely with

w

and

x

. When

y = 12

,

w = 8

, and

x = –5

,

z = –3

. What is the value of

k

, the constant of variation?

\newline

Write your answer in the form

A

or

\frac {(A)}{(B)}

, where

A

and

B

are constants or variable expressions.

\newline

____

Define Relationship Equation: The relationship between $z$ , $y$ , $w$ , and $x$ can be expressed as $z = k \cdot \left(\frac{y}{w \cdot x}\right)$ , where $k$ is the constant of variation we need to find.
Substitute Given Values: Given the values $y = 12$ , $w = 8$ , $x = -5$ , and $z = -3$ , we can substitute them into the equation to find $k$ : $-3 = k \times (12 / (8 \times -5))$ .
Simplify Denominator: Simplify the denominator of the fraction on the right side of the equation: $-3 = k \times \left(\frac{12}{-40}\right)$ .
Simplify Fraction: Simplify the fraction on the right side of the equation: $-3 = k \times \left(-\frac{12}{40}\right)$ .
Divide by Greatest Common Divisor: Further simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is $4$ : $-3 = k \times (-12 / 40) = k \times (-3 / 10)$ .
Multiply to Solve for $k$ : To solve for $k$ , multiply both sides of the equation by $-\frac{10}{3}$ : $k = (-3) \times (-\frac{10}{3})$ .
Find Value of k: Simplify the right side of the equation to find the value of $k$ : $k = 10$ .

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