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 sum of y and w.

Verified step by step guidance

1. Start by understanding the relationship: 'x varies directly as z and inversely as the sum of y and w'. This means x is proportional to z and inversely proportional to (y + w).
2. Write the equation for the direct and inverse variation: \( x = k \frac{z}{y + w} \), where \( k \) is the constant of proportionality.
3. To solve for \( y \), first multiply both sides by \( y + w \) to eliminate the fraction: \( x(y + w) = kz \).
4. Next, divide both sides by \( x \) to isolate \( y + w \): \( y + w = \frac{kz}{x} \).
5. Finally, solve for \( y \) by subtracting \( w \) from both sides: \( y = \frac{kz}{x} - w \).

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 can be expressed as x = k * z, where k is a constant. This means that as z increases, x increases proportionally, and vice versa.

Inverse Variation

Inverse variation occurs when one variable increases while another decreases, maintaining a constant product. For the given relationship, x varies inversely as the sum of y and w, which can be expressed as x = k / (y + w). This indicates that as the sum of y and w increases, x decreases.

Combining Direct and Inverse Variation

When a variable varies directly and inversely with respect to others, the relationship can be combined into a single equation. In this case, we can express the relationship as x = k * z / (y + w). To solve for y, we would rearrange this equation to isolate y, demonstrating the interplay between direct and inverse variations.