Many situations are more complicated than a basic direct variation or inverse variation model. One variable often depends on multiple other variables. When a variable is dependent on the product or quotient of two or more variables, this is called **joint variation**. For example, the cost of busing students for each school trip varies with the number of students attending and the distance from the school. The variable *c*, cost, varies jointly with the number of students, *n*, and the distance, *d*.

### A General Note: Joint Variation

Joint variation occurs when a variable varies directly or inversely with multiple variables.

For instance, if *x* varies directly with both *y* and *z*, we have *x *= *kyz*. If *x* varies directly with *y* and inversely with *z*, we have [latex]x=\frac{ky}{z}[/latex]. Notice that we only use one constant in a joint variation equation.

### Example 4: Solving Problems Involving Joint Variation

A quantity *x* varies directly with the square of *y* and inversely with the cube root of *z*. If *x *= 6 when *y *= 2 and *z *= 8, find *x* when *y *= 1 and *z *= 27.

### Solution

Begin by writing an equation to show the relationship between the variables.

Substitute *x *= 6, *y *= 2, and *z *= 8 to find the value of the constant *k*.

Now we can substitute the value of the constant into the equation for the relationship.

To find *x* when *y *= 1 and *z *= 27, we will substitute values for *y* and *z* into our equation.

### Try It 3

*x* varies directly with the square of *y* and inversely with *z*. If *x *= 40 when *y *= 4 and *z *= 2, find *x* when *y *= 10 and *z *= 25.