To determine the equation that describes the joint variation, follow these steps:
1. Understand Joint Variation: When a variable [tex]\( z \)[/tex] varies jointly with two other variables [tex]\( x \)[/tex] and [tex]\( y \)[/tex], it means that [tex]\( z \)[/tex] is proportional to the product of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. In mathematical terms, we can express this relationship as:
[tex]\[
z = k \cdot x \cdot y
\][/tex]
where [tex]\( k \)[/tex] is the constant of proportionality.
2. Substitute Given Values: We're given specific values for [tex]\( z \)[/tex], [tex]\( x \)[/tex], and [tex]\( y \)[/tex]:
[tex]\[
z = -75, \quad x = 3, \quad y = -5
\][/tex]
3. Find the Constant of Proportionality ([tex]\( k \)[/tex]): Substitute the given values into the joint variation equation:
[tex]\[
-75 = k \cdot 3 \cdot (-5)
\][/tex]
4. Simplify the Equation: Calculate the product of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
3 \cdot (-5) = -15
\][/tex]
So the equation simplifies to:
[tex]\[
-75 = k \cdot (-15)
\][/tex]
5. Solve for [tex]\( k \)[/tex]: Isolate [tex]\( k \)[/tex] by dividing both sides of the equation by [tex]\(-15\)[/tex]:
[tex]\[
k = \frac{-75}{-15}
\][/tex]
6. Calculate the Value of [tex]\( k \)[/tex]:
[tex]\[
k = 5
\][/tex]
7. Write the Final Equation: Substitute the value of [tex]\( k \)[/tex] back into the original joint variation equation:
[tex]\[
z = 5 \cdot x \cdot y
\][/tex]
Hence, the equation that describes the joint variation is:
[tex]\[
z = 5xy
\][/tex]
