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Ch. 5 - Systems and Matrices
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 5 - Systems and MatricesProblem 1
Chapter 6, Problem 1

Answer each question. By what expression should we multiply each side of 5/((3x(2x + 1)) = A/(3x) + B/(2x + 1) so that there are no fractions in the equation?

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1
Identify the denominators in the equation: the denominators are \$3x\( and \)(2x + 1)$.
To eliminate the fractions, multiply both sides of the equation by the least common denominator (LCD) of all the denominators involved.
The LCD is the product of the distinct factors in the denominators, which is \(3x(2x + 1)\).
Multiply each term on both sides of the equation by \(3x(2x + 1)\) to clear the fractions.
After multiplying, the equation will no longer have fractions, allowing you to work with a polynomial equation.

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Key Concepts

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

Least Common Denominator (LCD)

The least common denominator is the smallest expression that all denominators in a rational equation can divide into without leaving a remainder. Multiplying both sides of an equation by the LCD eliminates fractions, simplifying the equation for easier solving.
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Rationalizing Denominators

Partial Fraction Decomposition

Partial fraction decomposition breaks a complex rational expression into simpler fractions with simpler denominators. Understanding this helps identify the denominators involved and guides the process of clearing fractions by multiplying through by the LCD.
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Decomposition of Functions

Multiplying Equations to Clear Fractions

Multiplying both sides of an equation by an appropriate expression removes denominators, converting the equation into a polynomial form. This step is essential to avoid fractions and solve for variables more straightforwardly.
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Solving Linear Equations with Fractions
Related Practice
Textbook Question

What is the value of ∣40−20∣\(\left\)| \(\begin{matrix}\) 4 & 0 \\ -2 & 0 \(\end{matrix}\) \(\right\)|?

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Textbook Question

How many rows and how many columns does this matrix have? What is its dimension?

[−2580113−69]\(\left\)[ \(\begin{matrix}\) -2 & 5 & 8 & 0 \\ 1 & 13 & -6 & 9 \(\end{matrix}\) \(\right\)]

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Textbook Question

Answer each of the following. When appropriate, fill in the blank to correctly complete the sentence. The following nonlinear system has two solutions, one of which is (3,____).

x + y = 7

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Textbook Question

Answer each question. What is the product of [64−18]\(\left\)[ \(\begin{matrix}\) 6 & 4 \\ -1 & 8 \(\end{matrix}\) \(\right\)] and I2 (in either order)?

Textbook Question

Fill in the blank(s) to correctly complete each sentence. For the following statement to be true, the value of x must be ____, and the value of y must be ____.

[3−651]=[x+1−65y+1]\(\left\)[ \(\begin{matrix}\) 3 & -6 \\ 5 & 1 \(\end{matrix}\) \(\right\)] = \(\left\)[ \(\begin{matrix}\) x + 1 & -6 \\ 5 & y + 1 \(\end{matrix}\) \(\right\)]

Textbook Question

Match each system of inequalities with the correct graph from choices A–D. x ≥ 5 y ≤ -3