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

Verified step by step guidance

1. Identify the given variation relationship: "x varies directly as the cube of z and inversely as y." This means x is proportional to \( z^3 \) and inversely proportional to \( y \).
2. Write the variation equation using a constant of proportionality \( k \): \[ x = \frac{k \cdot z^3}{y} \]
3. To solve for \( y \), multiply both sides of the equation by \( y \) to get rid of the denominator: \[ x \cdot y = k \cdot z^3 \]
4. Next, isolate \( y \) by dividing both sides by \( x \): \[ y = \frac{k \cdot z^3}{x} \]
5. The equation is now solved for \( y \) in terms of \( x \), \( z \), and the constant \( k \).

Key Concepts

Direct Variation

Direct variation describes a relationship where one variable is proportional to another. If x varies directly as the cube of z, it means x = k * z³ for some constant k, indicating that as z increases, x changes proportionally to z cubed.

Inverse Variation

Inverse variation means one variable changes in the opposite way to another. Here, x varies inversely as y, so x = k / y, implying that as y increases, x decreases proportionally, and vice versa.

Solving for a Variable in an Equation

Solving for y involves manipulating the equation algebraically to isolate y on one side. This requires understanding how to rearrange terms, use inverse operations, and simplify expressions to express y explicitly in terms of x and z.