In Exercises 11–20, 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 relationship described: x varies jointly as z and the difference between y and w. This can be expressed as x = k(z(y - w)), where k is the constant of variation.
2. Rearrange the equation to solve for y. Start by isolating the term involving y: x = kzy - kwz.
3. Isolate the term with y on one side: kzy = x + kwz.
4. Divide both sides by kz to solve for y: y = \frac{x + kwz}{kz}.
5. This is the equation expressing y in terms of x, z, w, and k. You can use this equation to find y given values for x, z, w, and k.

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, x varies jointly as z and the difference between y and w, meaning that x can be expressed as a constant multiplied by both z and (y - w). Understanding this relationship is crucial for setting up the correct equation.

Setting Up Equations

To express the relationship described in the problem, we need to formulate an equation based on the joint variation. This involves identifying the constant of variation (k) and writing the equation as x = k * z * (y - w). This step is essential for translating the verbal description into a mathematical expression that can be manipulated.

Solving for y

Once the equation is established, solving for y involves isolating y on one side of the equation. This typically requires algebraic manipulation, such as distributing, combining like terms, and using inverse operations. Mastery of these techniques is necessary to find the value of y in terms of the other variables and the constant.