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

Verified step by step guidance

1. Identify the phrase 'x varies jointly as z and the difference between 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 containing \(y\). Divide both sides of the equation by \(kz\): \(\frac{x}{kz} = y - w\).
4. Next, solve for \(y\) by adding \(w\) to both sides: \(y = \frac{x}{kz} + w\).
5. The equation is now expressed with \(y\) as the subject: \(y = \frac{x}{kz} + w\). This shows \(y\) 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 difference between 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 'x varies jointly as z and (y - w)' helps set up the equation x = k * z * (y - w), which models the given relationship.

Solving Equations for a Specific Variable

Solving for y means isolating y on one side of the equation. Starting from x = k * z * (y - w), you divide both sides by k*z and then add w to isolate y, resulting in y = (x / (k*z)) + w.