Problem 60 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1 - 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: \(\frac{x}{xy} + \frac{y}{xy}\).

Step 2 - Combine the fractions

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

Step 3 - 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 \(\frac{1}{xy}\).

Key Concepts

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.