Write an equation that expresses each relationship. Then solve the equation for y. x varies jointly as y and z and inversely as the square of w.

Verified step by step guidance

1. Identify the type of variation described: "x varies jointly as y and z" means x is proportional to the product of y and z, so we write \(x \propto y \cdot z\).
2. The phrase "inversely as the square of w" means x is inversely proportional to \(w^2\), so we include this as \(x \propto \frac{1}{w^2}\).
3. Combine both parts to write the joint and inverse variation as an equation with a constant of proportionality \(k\): \(x = k \cdot \frac{y \cdot z}{w^2}\).
4. To solve for \(y\), multiply both sides of the equation by \(w^2\) to get rid of the denominator: \(x \cdot w^2 = k \cdot y \cdot z\).
5. Finally, isolate \(y\) by dividing both sides by \(k \cdot z\): \(y = \frac{x \cdot w^2}{k \cdot z}\).

Key Concepts

Joint Variation

Joint variation describes a relationship where one variable varies directly as the product of two or more other variables. In this case, x varies jointly as y and z means x is proportional to y multiplied by z, expressed as x = kyz for some constant k.

Inverse Variation

Inverse variation means one variable varies inversely as another variable or its power. Here, x varies inversely as the square of w, meaning x is proportional to 1 divided by w squared, or x = k / w², where k is a constant.

Solving for a Variable in an Equation

Solving for y involves isolating y on one side of the equation. After writing the variation equation, algebraic manipulation such as multiplication, division, or taking roots is used to express y explicitly in terms of x, z, w, and constants.