# If z is directly proportional to the product of x and y and if z is 10 when x is 4 and y is 5,...

## Question:

If z is directly proportional to the product of x and y and if z is 10 when x is 4 and y is 5, then x, y, and z are related by what equation?

## Joint Variation

Joint variation gives a relationship between one dependent variable and two or more independent variables, whereby the dependent variable varies directly with the independent variables when all the other variables are held constant.

If the variable {eq}z {/eq} is directly proportional to the product of {eq}x {/eq} and {eq}y {/eq}, then we can write this as:

• {eq}z \propto xy {/eq}

Removing the proportionality sign and adding a proportionality constant, we get:

• {eq}z = kxy {/eq}

Given that {eq}z = 10 {/eq} when {eq}x = 4 {/eq} and {eq}y = 5 {/eq}, then:

• {eq}10 = k\times 4\times 5 {/eq}
• {eq}10 = 20k {/eq}

Solving for k:

• {eq}k = \dfrac{10}{20} = 0.5 {/eq}

Thus, the equation showing the relationship between the three variables is:

• {eq}\boxed{\color{blue}{y = 0.5xy}} {/eq} 