Textbook Question
In Exercises 1–18, solve each system by the substitution method.
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Ch. 5 - Systems of Equations and Inequalities
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 6, Problem 9
Write the partial fraction decomposition of each rational expression. x/(x-2)(x-3)
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Identify the form of the rational expression. Since the denominator is a product of two distinct linear factors, \((x-2)(x-3)\), the partial fraction decomposition will have the form: \(\frac{A}{x-2} + \frac{B}{x-3}\), where \(A\) and \(B\) are constants to be determined.
Write the equation equating the original fraction to the sum of the partial fractions: \(\frac{x}{(x-2)(x-3)} = \frac{A}{x-2} + \frac{B}{x-3}\).
Multiply both sides of the equation by the common denominator \((x-2)(x-3)\) to clear the denominators: \(x = A(x-3) + B(x-2)\).
Expand the right side: \(x = A x - 3A + B x - 2B\).
Group like terms and set up a system of equations by equating coefficients of corresponding powers of \(x\): \(x = (A + B) x + (-3A - 2B)\). This gives the system: \(A + B = 1\) (coefficient of \(x\)) and \(-3A - 2B = 0\) (constant term).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Rational Expressions
A rational expression is a fraction where both the numerator and denominator are polynomials. Understanding how to manipulate these expressions is essential for simplifying, factoring, and decomposing them into partial fractions.
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Rationalizing Denominators
Partial Fraction Decomposition
Partial fraction decomposition involves expressing a complex rational expression as a sum of simpler fractions with linear or irreducible quadratic denominators. This technique is useful for integration and solving equations involving rational expressions.
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Factoring Denominators
Factoring the denominator into linear or irreducible quadratic factors is a crucial step before decomposition. It helps identify the form of the partial fractions and determines the structure of the numerators in the decomposition.
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Related Practice
Textbook Question
An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of x and y for which the maximum occurs.
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Textbook Question
A chemist needs to mix a solution that is 34% silver nitrate with one that is 4% silver nitrate to obtain 100 milliliters of a mixture that is 7% silver nitrate. How many milliliters of each of the solutions must be used?
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Textbook Question
In Exercises 5–18, solve each system by the substitution method.
Textbook Question
Solve each system in Exercises 5–18.
Textbook Question
In Exercises 1–8, write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants. (7x2 -9x+3)/(x2+7)2