Write an equation that expresses each relationship. Then solve the equation for y. x varies directly as z and inversely as the sum of 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 "and inversely as the sum of y and w" means x is inversely proportional to \((y + w)\), so we include this in the equation as \(x = \frac{k \cdot z}{y + w}\).
3. Write the full equation expressing the relationship: \(x = \frac{k \cdot z}{y + w}\), 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, subtract \(w\) from both sides to solve for \(y\): \(y = \frac{k \cdot z}{x} - w\).

Key Concepts

Direct Variation

Direct variation describes a relationship where one variable is proportional to another. If x varies directly as z, it means x = k * z for some constant k. This implies that as z increases, x increases proportionally, and vice versa.

Inverse Variation

Inverse variation means one variable changes in the opposite way to another. If x varies inversely as a quantity, then x = k / (that quantity). Here, as the denominator increases, x decreases, showing an inverse proportionality.

Forming and Solving Equations for a Variable

To solve for y, first write the equation expressing the given variation relationships. Then, isolate y by algebraic manipulation, such as multiplying both sides, combining like terms, and using inverse operations to express y explicitly.