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Chapter 7: Problem 81

Multiply or divide as indicated. $$ \frac{12 x-10 y}{3 x+2 y} \cdot \frac{6 x+4 y}{10 y-12 x} $$

Short Answer

Expert verified
\( -1 \)

Step by step solution

01

Simplify Each Fraction

First, observe each fraction to see if they can be simplified. The fractions given are \( \frac{12x - 10y}{3x + 2y} \) and \( \frac{6x + 4y}{10y - 12x} \).
02

Factor Expressions

Factor out the greatest common factor (GCF) from each numerator and denominator. \( 12x - 10y = 2(6x - 5y), \) \( 3x + 2y = 3x + 2y, \) \( 6x + 4y = 2(3x + 2y), \) \( 10y - 12x = 2(5y - 6x) \).
03

Rewrite the Fractions

Rewrite each fraction with the factored form. \( \frac{2(6x - 5y)}{3x + 2y} \cdot \frac{2(3x + 2y)}{2(5y - 6x)} \).
04

Reduce the Fraction

Cancel out common factors in the numerators and denominators. \( \frac{2(6x - 5y)}{3x + 2y} \cdot \frac{2(3x + 2y)}{2(5y - 6x)} = \frac{2 \times 2(6x - 5y) \times (3x + 2y)}{(3x + 2y) \times 2(5y - 6x)} \ \rightarrow \) Cancel out the \( 3x + 2y \) and the factor of \( 2 \): = \( \frac{2(6x - 5y)}{5y - 6x} \).
05

Identify Complex Factors

Recognize that \(5y - 6x\) is the negative of \(6x - 5y\). Thus, \( \frac{6x - 5y}{5y - 6x} = -1 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

factoring expressions
Factoring is the process of breaking down an expression into a product of simpler factors. This is helpful in simplifying algebraic fractions. To factor an expression, look for the greatest common factor (GCF) first.
For example, in the expression \(12x - 10y\), the GCF is 2. So, we can factor it as \(2(6x - 5y)\).
Similarly, let's factor the expression \(6x + 4y\). The GCF here is 2, so it factors into \(2(3x + 2y)\).
Factoring expressions makes it easier to simplify and solve algebraic problems.
simplifying fractions
Simplifying fractions involves reducing them to their simplest form. Simplification occurs when the numerator and the denominator share a common factor that can be canceled out.
Take the fraction \(\frac{2(6x - 5y)}{3x + 2y}\). If the numerator and the denominator have common factors, these can be canceled.
Observe that in the problem, we can simplify: \(\frac{2(6x - 5y)}{3x + 2y} \cdot \frac{2(3x + 2y)}{2(5y - 6x)}\).
This results in canceling the common terms \((3x + 2y)\) and the factor 2.
The simplified form of the expression will be: \(\frac{2(6x - 5y)}{5y - 6x}\).
greatest common factor
The greatest common factor (GCF) is the largest number or expression that divides two or more numbers or expressions without any remainder.
For example, for the terms \(12x\) and \(10y\), the GCF is 2 because it is the highest number that divides both 12 and 10 evenly.
When factoring expressions like \(12x - 10y\), always begin by identifying the GCF. Here, \(12x - 10y\) can be factored using the GCF 2 to get \(2(6x - 5y)\).
Knowing how to find the GCF is crucial for simplifying algebraic fractions and making the problem more manageable.

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Most popular questions from this chapter

Multiply or divide as indicated. $$ \frac{3 k^{2}+17 k p+10 p^{2}}{6 k^{2}+13 k p-5 p^{2}} \div \frac{6 k^{2}+k p-2 p^{2}}{6 k^{2}-5 k p+p^{2}} $$

Add or subtract as indicated. Write all answers in lowest terms. $$ \frac{3 x}{x+1}+\frac{4}{x-1}-\frac{6}{x^{2}-1} $$

Solve each problem. See Examples \(1-7\) In \(2007,51.2 \%\) of the homes in the United States used natural gas as the primary heating fuel. (Source: U.S. Census Bureau.) The volume of gas varies inversely as the pressure and directly as the temperature. (Temperature must be measured in Kelvin (K), a unit of measurement used in physics.) If a certain gas occupies a volume of \(1.3 \mathrm{L}\) at \(300 \mathrm{K}\) and a pressure of 18 newtons, find the volume at \(340 \mathrm{K}\) and a pressure of 24 newtons.

Add or subtract as indicated. Write all answers in lowest terms. $$ \frac{3}{(p-2)^{2}}-\frac{5}{p-2}+4 $$

Add or subtract as indicated. Write all answers in lowest terms. $$ \frac{11 x-13}{2 x-3}-\frac{3 x-1}{2 x-3} $$
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