Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

1. Equations & Inequalities

Rational Equations

College Algebra

1. Equations & Inequalities

Rational Equations

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2:17 minutes

Problem 17

Textbook Question

In Exercises 11–20, write an equation that expresses each relationship. Then solve the equation for y.x varies jointly as z and the sum of y and w.

Verified step by step guidance

Identify the relationship described: x varies jointly as z and the sum of y and w. This means x is directly proportional to the product of z and the sum (y + w).

Write the equation based on the joint variation relationship: x = k(z(y + w)), where k is the constant of proportionality.

Rearrange the equation to solve for y. Start by isolating the term containing y: x = kz(y + w).

Divide both sides of the equation by kz to isolate (y + w): \frac{x}{kz} = y + w.

Finally, solve for y by subtracting w from both sides: y = \frac{x}{kz} - w.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 sum of y and w, meaning that x can be expressed as a constant multiplied by z and (y + w). Understanding this relationship is crucial for forming the correct equation.

Formulating Equations

To express the relationship mathematically, we need to formulate an equation based on the given variation. For joint variation, the equation takes the form x = k * z * (y + w), where k is the constant of variation. This step is essential for translating the verbal description into a solvable mathematical expression.

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.