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

Verified step by step guidance

1. Identify the type of variation described: "x varies directly as z" means \(x\) is proportional to \(z\), so we can write \(x = k \cdot z\) for some constant \(k\).
2. The phrase "inversely as the difference between y and w" means \(x\) is inversely proportional to \((y - w)\), so we include this as a denominator: \(x = \frac{k \cdot z}{y - w}\).
3. Write the combined variation equation: \(x = \frac{k \cdot z}{y - w}\), where \(k\) is the constant of proportionality.
4. To solve for \(y\), start by multiplying both sides of the equation by \((y - w)\) to eliminate the denominator: \(x(y - w) = k \cdot z\).
5. Next, divide both sides by \(x\) to isolate \((y - w)\): \(y - w = \frac{k \cdot z}{x}\). Finally, add \(w\) to both sides to solve for \(y\): \(y = w + \frac{k \cdot z}{x}\).

Key Concepts

Direct Variation

Direct variation describes a relationship where one variable increases or decreases proportionally with another. If x varies directly as z, it means x = k * z for some constant k. This concept helps set up equations involving proportional relationships.

Inverse Variation

Inverse variation occurs when one variable increases as another decreases, such that their product is constant. If x varies inversely as (y - w), then x = k / (y - w) for some constant k. This helps model relationships where variables are inversely related.

Solving for a Variable in an Equation

Solving for y means isolating y on one side of the equation. This involves algebraic manipulation such as multiplying both sides, adding or subtracting terms, and simplifying expressions. Mastery of these steps is essential to express y explicitly.