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Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
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Tutti i libri di testoBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblema 19
Capitolo 4, Problema 19

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.

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Identify the type of variation described: "x varies directly as z" means \(x\) is proportional to \(z\), so we can write \(x = k \cdot z\) for some constant \(k\).
The phrase "inversely as the difference between y and w" means \(x\) is inversely proportional to \((y - w)\), so we include this as a denominator: \(x = \frac{k \cdot z}{y - w}\).
Write the combined variation equation: \(x = \frac{k \cdot z}{y - w}\), where \(k\) is the constant of proportionality.
To solve for \(y\), start by multiplying both sides of the equation by \((y - w)\) to eliminate the denominator: \(x(y - w) = k \cdot z\).
Next, divide both sides by \(x\) to isolate \((y - w)\): \(y - w = \frac{k \cdot z}{x}\). Finally, add \(w\) to both sides to solve for \(y\): \(y = w + \frac{k \cdot z}{x}\).

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Direct Variation

Direct variation describes a relationship where one variable increases or decreases proportionally with another. If x varies directly as z, it means x = k * z for some constant k. This concept helps set up equations involving proportional relationships.
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Inverse Variation

Inverse variation occurs when one variable increases as another decreases, such that their product is constant. If x varies inversely as (y - w), then x = k / (y - w) for some constant k. This helps model relationships where variables are inversely related.
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Solving for a Variable in an Equation

Solving for y means isolating y on one side of the equation. This involves algebraic manipulation such as multiplying both sides, adding or subtracting terms, and simplifying expressions. Mastery of these steps is essential to express y explicitly.
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Equations with Two Variables
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