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

Verified step by step guidance

1. Understand the phrase "x varies jointly as y and z" means that x is directly proportional to the product of y and z. This can be expressed as an equation involving a constant of proportionality, say k.
2. Write the joint variation equation as \(x = k \cdot y \cdot z\), where \(k\) is a constant.
3. To solve the equation for \(y\), isolate \(y\) on one side. Start by dividing both sides of the equation by \(kz\) (assuming \(k \neq 0\) and \(z \neq 0\)):
4. \[ y = \frac{x}{kz} \]
5. This equation expresses \(y\) in terms of \(x\), \(k\), and \(z\). If you have specific values for \(x\), \(k\), and \(z\), you can substitute them to find \(y\).

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 case, x varies jointly as y and z means x = kyz, where k is the constant of proportionality.

Formulating Equations from Word Problems

Translating a verbal relationship into an algebraic equation involves identifying variables and their relationships. Here, recognizing 'x varies jointly as y and z' helps write the equation x = kyz, setting the foundation for solving for a specific variable.

Solving for a Variable

Solving for y means isolating y on one side of the equation. Starting from x = kyz, divide both sides by kz (assuming k and z are nonzero) to get y = x / (kz), which expresses y explicitly in terms of x, k, and z.