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

Verified step by step guidance

1. Identify the phrase 'x varies jointly as z and the sum of y and w.' This means x is proportional to both z and (y + w) multiplied together.
2. Write the joint variation equation as: \(x = k \cdot z \cdot (y + w)\), where \(k\) is the constant of proportionality.
3. To solve for \(y\), start by isolating the term \((y + w)\): divide both sides of the equation by \(k \cdot z\) to get \(\frac{x}{k \cdot z} = y + w\).
4. Next, isolate \(y\) by subtracting \(w\) from both sides: \(y = \frac{x}{k \cdot z} - w\).
5. The equation is now solved for \(y\) in terms of \(x\), \(z\), \(w\), and the constant \(k\).

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 problem, x varies jointly as z and the sum of y and w, meaning x = k * z * (y + w) for some constant k.

Formulating Equations from Word Problems

Translating verbal descriptions into algebraic equations involves identifying variables and their relationships. Here, recognizing that 'x varies jointly as z and the sum of y and w' leads to an equation involving multiplication of z and (y + w) with a constant.

Solving Equations for a Specific Variable

Solving for y means isolating y on one side of the equation. This often involves algebraic manipulation such as division, subtraction, and factoring to rewrite the equation explicitly in terms of y.