In Exercises 11–20, 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. Identify the type of variation described in the problem. In this case, 'x varies jointly as y and z' indicates a joint variation, which can be represented by the equation x = kyz, where k is the constant of variation.
2. Rewrite the equation to isolate y. Start by dividing both sides of the equation by z to get y on one side: y = \frac{x}{kz}.
3. To solve for y explicitly, express k in terms of x, y, and z if the values are known, or leave it as k if they are not specified.
4. Check the units or dimensions in the problem to ensure consistency across the equation, especially if values for x, y, z, or k are given.
5. Use the derived formula y = \frac{x}{kz} to find y when x and z are known, or to analyze how changes in x and z affect y.

Key Concepts

Joint Variation

Joint variation occurs when a variable is directly proportional to the product of two or more other variables. In this case, if x varies jointly as y and z, it means that x can be expressed as a constant multiplied by both y and z. The general form of the equation is x = k * y * z, where k is the constant of variation.

Solving for a Variable

Solving for a variable involves rearranging an equation to isolate the desired variable on one side. In this context, after establishing the equation from the joint variation, we need to manipulate it algebraically to express y in terms of x and z. This often requires dividing both sides of the equation by the other variables.

Constants of Variation

The constant of variation (k) is a specific value that relates the variables in a joint variation equation. It represents the proportionality factor that remains constant for given values of x, y, and z. Understanding how to determine or use this constant is crucial for accurately expressing and solving equations involving joint variation.