How does z vary directly with the sum of the squares of x and y?

When we say that a variable z varies directly with another variable or a combination of variables, it means that there exists a constant k such that:

z = k * (x² + y²)

In this case, we understand that z is proportional to the sum of the squares of x and y. This relationship implies that if the sum of the squares of x and y increases, z will also increase, provided that the constant k remains unchanged.

To illustrate this concept, let’s say we have a specific value for k. For example, if k = 2 and we substitute x = 3 and y = 4, we can calculate z:

• First, compute the squares: x² = 3² = 9
• y² = 4² = 16

Now, we sum these squares:

x² + y² = 9 + 16 = 25

Finally, we can find z:

z = 2 * 25 = 50

This example demonstrates that the relationship holds: as the squares of the inputs increase, even with a constant factor, so does z. Thus, z varies directly as specified by the equation z = k * (x² + y²).