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 root 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 \(x = k \cdot y \cdot z\) where \(k\) is a constant of proportionality.
2. Since x also varies inversely as the square root of w, include this inverse relationship by dividing by \(\sqrt{w}\). The equation becomes \(x = \frac{k \cdot y \cdot z}{\sqrt{w}}\).
3. Write the full equation expressing the relationship: \(x = \frac{k \cdot y \cdot z}{\sqrt{w}}\).
4. To solve for y, multiply both sides of the equation by \(\sqrt{w}\) to get rid of the denominator: \(x \cdot \sqrt{w} = k \cdot y \cdot z\).
5. Finally, isolate y by dividing both sides by \(k \cdot z\): \(y = \frac{x \cdot \sqrt{w}}{k \cdot z}\).

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 the product y * z, expressed as x = k * y * z, where k is a constant.

Inverse Variation

Inverse variation means one variable varies inversely as another, so as one increases, the other decreases proportionally. Here, x varies inversely as the square root of w, meaning x is proportional to 1 divided by √w, or x = k / √w.

Solving for a Variable in an Equation

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