How can we express the relationship between y, x, and z when it’s given that y varies directly with x and inversely with z, and specifically, if y is 18 when x is 15 and z is 5?

To express the relationship among the variables, we will start by understanding the concept of direct and inverse variation. When we say that y varies directly with x, it means that if x increases, y increases, and similarly, if x decreases, y decreases. In contrast, y varies inversely with z means that if z increases, y decreases, and vice-versa.

We can combine these two relationships into a single equation. The general form for a variable that varies directly with one variable and inversely with another is:

y = k * (x / z)

In this equation, k is a constant of proportionality that we need to determine using the provided conditions.

Given that y = 18 when x = 15 and z = 5, we can substitute these values into our equation to solve for k:

18 = k * (15 / 5)

Now simplify the expression:

18 = k * 3

Next, we can solve for k:

k = 18 / 3

k = 6

Now we can substitute the value of k back into the original equation:

y = 6 * (x / z)

Therefore, the equation that describes the relationship where y varies directly with x and inversely with z is:

y = 6 * (x / z)

This equation shows how y changes based on the values of x and z. You can use it to calculate y for any values of x and z within the applicable context.