In Exercises 11–20, write an equation that expresses each relationship. Then solve the equation for y. x varies directly as z and inversely as the difference between y and w.

Verified step by step guidance

1. Identify the type of variation in the problem. Since x varies directly with z and inversely with the difference between y and w, you can express this relationship using the formula for combined variation: x = k \frac{z}{y-w}, where k is the constant of proportionality.
2. Isolate y in the equation to solve for it in terms of x, z, w, and k. Start by multiplying both sides of the equation by (y-w) to eliminate the denominator: x(y-w) = kz.
3. Expand the left side of the equation: xy - xw = kz.
4. Rearrange the equation to isolate terms involving y on one side: xy = kz + xw.
5. Finally, solve for y by dividing both sides of the equation by x: y = \frac{kz + xw}{x}. This expresses y in terms of x, z, w, and k.

Key Concepts

Direct Variation

Direct variation describes a relationship where one variable is a constant multiple of another. In this case, if x varies directly as z, it means that x = k * z for some constant k. This concept is fundamental in understanding how changes in one variable directly affect another.

Inverse Variation

Inverse variation occurs when one variable increases while another decreases, maintaining a constant product. Here, x varies inversely as the difference between y and w, which can be expressed as x = k / (y - w). This relationship is crucial for establishing how the variables interact in the equation.

Solving for a Variable

Solving for a variable involves rearranging an equation to isolate the variable of interest. In this context, after establishing the relationship between x, z, y, and w, the goal is to manipulate the equation to express y in terms of the other variables. This skill is essential for finding specific values and understanding the relationships between the variables.