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Chapter 7: Problem 26

Perform each indicated operation. Simplify if possible. See Examples I through 7. $$ \frac{-y+1}{y}-\frac{2 y-5}{3 y} $$

Short Answer

Expert verified
\( \frac{-5y+8}{3y} \)

Step by step solution

01

Find a Common Denominator

To subtract these fractions, we need a common denominator. The denominators are \( y \) and \( 3y \). The least common denominator (LCD) is \( 3y \), as it is the least common multiple of the two denominators.
02

Rewrite the First Fraction

The first fraction is \( \frac{-y+1}{y} \). To express it with the common denominator \( 3y \), multiply both the numerator and the denominator by 3, resulting in \( \frac{3(-y+1)}{3y} = \frac{-3y+3}{3y} \).
03

Rewrite the Second Fraction

The second fraction \( \frac{2y-5}{3y} \) already has the common denominator \( 3y \). Thus, we do not need to modify it.
04

Subtract the Fractions

Now, subtract the second fraction from the first one:\[\frac{-3y+3}{3y} - \frac{2y-5}{3y} = \frac{(-3y+3) - (2y-5)}{3y} = \frac{-3y+3 - 2y + 5}{3y} \].
05

Simplify the Numerator

Combine like terms in the numerator:\[-3y - 2y + 3 + 5 = -5y + 8 \].So the expression becomes:\[\frac{-5y+8}{3y} \].
06

Final Simplification

The fraction \( \frac{-5y+8}{3y} \) cannot be simplified further since there are no common factors in the numerator and the denominator. This is our final simplified expression.

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Key Concepts

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

Common Denominator
When dealing with fractions, the common denominator is essential, especially for arithmetic operations like addition and subtraction. A common denominator is a shared multiple of the denominators of two or more fractions. To subtract fractions, we first need to convert them into equivalent fractions with the same denominator.
  • Determine the denominators of the fractions: In our problem, the denominators are \( y \) and \( 3y \).
  • Find the least common denominator (LCD): For the denominators \( y \) and \( 3y \), \( 3y \) is the LCD because it is the smallest expression divisible by both \( y \) and \( 3y \).
  • Rewrite each fraction with the newly found common denominator: Multiply the numerator and denominator of each fraction by the necessary factors.
Once you have a common denominator, subtracting becomes straightforward because you're essentially combining like terms.
Subtracting Fractions
Subtracting fractions sounds tricky, but once you have a common denominator, it's a straightforward process. Let鈥檚 break it down:First, rewrite each fraction so they share the same denominator. In our example:- The first fraction \( \frac{-y+1}{y} \) is rewritten as \( \frac{-3y+3}{3y} \).- The second fraction \( \frac{2y-5}{3y} \) is already expressed with the common denominator. Now, subtract the numerators while keeping the common denominator:- Combine the numerators: Take the numerator of the first fraction, \(-3y + 3\), and subtract the numerator of the second fraction, \(2y - 5\).- Be careful with signs: Subtract \((2y - 5)\) by distributing the negative sign inside the parentheses.The result of subtracting the numerators is \(-3y + 3 - 2y + 5 = -5y + 8\). This gives us the fraction \(\frac{-5y+8}{3y}\) with a simplified numerator.
Simplifying Expressions
Simplifying expressions is the icing on the cake after performing arithmetic operations on fractions. It is crucial because it makes the expression easier to understand and work with in later calculations.After you鈥檝e subtracted the fractions, you'll want to simplify the result. Here's how:- Once you've reached an expression like \(\frac{-5y+8}{3y}\), check the numerator and denominator for common factors.- Simplifying involves dividing both the numerator and the denominator by their greatest common factor. In this specific exercise, there are no common factors between \(-5y+8\) and \(3y\), meaning this is as simplified as it gets.Simplification helps in reducing the complexity of expressions and ensures that your results are as concise as possible. Always aim for the simplest form unless additional context suggests otherwise.

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Most popular questions from this chapter

Simplify each complex fraction. See Examples 1 and 2. $$ \frac{\frac{x+3}{x^{2}-9}}{1+\frac{1}{x-3}} $$

Perform each indicated operation. Explain when the LCD of the rational expressions in a sum is the product of the denominators.

One of the great algebraists of ancient times a man named Diophantus. Litle is known of his life other than the lived and worked in Alexandria. Some historians believe he lived during the first century of the Christian era, about the time of Nero. The only che to his personal life is the following epigram found in a collection called the Palatine A nthology. God granted him youth for a sixth of his life and added a twelfth part to this. He clothed his cheeks in down. He lit him the light of wedlock after a seventh part and five years after his marriage. He granted him a son. Alas, lateborn wretched child. After attaining the measure of half his father's life, cruel fate overtook him, thus leaving Diophantus during the last four years of his life only such consolation as the science of numbers. How old was Diophantus at his death?" We are looking for Diophantus' age when he died, so let x represent that age. If we sum the parts of his life, we should get the total age. Parts of his life \(\left\\{\begin{array}{l}{\frac{1}{6} x+\frac{1}{12} x \text { is the time of his youth. }} \\ {\frac{1}{7} x \text { is the time between his youth and when }} \\ {\text { he married. }} \\ {5 \text { years is the time between his marriage }} \\ {\text { and the birth of his son. }} \\\ {\frac{1}{2} x \text { is the time Diophantus had with his son. }} \\ {4 \text { years is the time between his son's death }} \\ {\text { and his own. }}\end{array}\right.\) The sum of these parts should equal Diophantus' age when he died. $$ \frac{1}{6} \cdot x+\frac{1}{12} \cdot x+\frac{1}{7} \cdot x+5+\frac{1}{2} \cdot x+4=x $$ How old was Diophantus when his son was born? How old was the son when he died?

\- Solve the following. See Examples I through 7. (Note: Some exercises can be modeled by equations without rational expressions.) A pilot can fly a DC-10 1365 miles against the wind in the same time as he can fly 1575 miles with the wind. If the speed of the plane in still air is 490 miles per hour, find the speed of the wind. (Source: Air Transport Association of America)

Solve. See the Concept Check in the Section. Which of the following are equivalent to \(\frac{\frac{a}{7}}{\frac{b}{13}} ?\) a. \(\frac{a}{7} \cdot \frac{b}{13}\) b. \(\frac{a}{7} \div \frac{b}{13}\) c. \(\frac{a}{7} \div \frac{13}{b}\) d. \(\frac{a}{7} \cdot \frac{13}{b}\)
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