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Chapter 7: Problem 60

For exercises \(25-68\), evaluate or simplify. $$ \frac{\frac{1}{y}+\frac{1}{x}}{y+x} $$

Short Answer

Expert verified
The simplified form is \(\frac{1}{xy}\).

Step by step solution

01

- Find a common denominator

To combine the fractions in the numerator \(\frac{1}{y}+\frac{1}{x}\), first find a common denominator. The least common multiple of y and x is xy. So, rewrite each fraction with this common denominator: \(\frac{x}{xy} + \frac{y}{xy}\).
02

- Combine the fractions

Now, add the two fractions together: \(\frac{x}{xy} + \frac{y}{xy} = \frac{x + y}{xy}\). The numerator becomes \(\frac{x + y}{xy}\).
03

- Simplify the entire expression

Rewrite the expression with the new numerator: \(\frac{\frac{x + y}{xy}}{y + x}\). Notice that \(\frac{x + y}{y + x}\) equals 1 since they are the same. Thus, the expression simplifies to \(\frac{1}{xy}\).

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Key Concepts

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

Common Denominator
A crucial concept for simplifying rational expressions is the common denominator. To add fractions, they must share the same denominator. This common denominator enables us to combine the fractions into a single fraction. For instance, in the exercise, the fractions \(\frac{1}{y} \) and \(\frac{1}{x} \) have different denominators. We need to find a common denominator to add these fractions effectively.
This common denominator is typically the least common multiple (LCM) of the denominators.
For the given fractions, the LCM of y and x is xy. Thus, we rewrite the fractions with respect to this common denominator.
Combining Fractions
Once you've established a common denominator, combining fractions becomes straightforward. Rewrite each fraction so they share this common denominator. For instance:
  • \(\frac{1}{y} \) becomes \(\frac{x}{xy} \)
  • \(\frac{1}{x} \) becomes \(\frac{y}{xy} \)
After this step, you can add the numerators while keeping the common denominator the same. This simplifies the expression:
\(\frac{x}{xy} + \frac{y}{xy} = \frac{x + y}{xy}\).
This new fraction combines the two original fractions efficiently.
Least Common Multiple
The concept of the least common multiple (LCM) is essential in rational expressions. It represents the smallest number that is a multiple of two or more numbers. In our exercise, the LCM serves as the common denominator. The denominators involved are y and x; their LCM is found by multiplying them together, giving us xy.
This step is fundamental because it paves the way for the fractions to be added or subtracted without modifying the values.
Algebraic Expressions
Algebraic expressions like \(\frac{\frac{1}{y}+\frac{1}{x}}{y+x}\) can be challenging but breaking them down step-by-step helps. In our simplified expression:
\(\frac{\frac{x + y}{xy}}{y + x}\)
we see that the numerator and the denominator share a common term (x + y). When the same term appears in both the numerator and the denominator, they can cancel each other out, simplifying the expression:
\(\frac{1}{xy}\)
Understanding how to manipulate and simplify algebraic expressions using such techniques is crucial for solving more complex problems.

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Most popular questions from this chapter

The height of a triangle is \(3 \mathrm{ft}\) more than the length of its base, and its area is \(54 \mathrm{ft}^{2}\). Use a quadratic equation to find the base and height of this triangle. \(\left(A=\frac{1}{2} b h .\right)\)

For exercises \(67-82\), use the five steps and a proportion. Find the number of adults used to create the ratio "four out of five." Four out of five adults now use the Internet. 184 million adults are online from their homes, offices, schools or other locations. (Source: www.harrisinteractive.com, Nov. 17, 2008)

The relationship of \(x\) and \(y\) is a direct variation. When \(x=1, y=6\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this direct variation. c. Find \(y\) when \(x=4\), d. Use slope-intercept graphing to graph this equation. e. Use the graph to find \(y\) when \(x=2\).

For exercises \(67-82\), use the five steps and a proportion. In \(2010,3.5\) per 100,000 full-time equivalent workers were killed on the job with a total of 547 workers killed on the job. Find the number of full-time equivalent workers used to create this ratio. Round to the nearest whole number. (Source: www.osha.gov)

For exercises 1-10, (a) solve. (b) check. $$ \frac{1}{6} w+\frac{23}{8}=-3 $$
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