Problem 49 - Elementary Algebra

Question

For Problems 41-60, simplify each of the complex fractions. $$ \frac{\frac{2}{x}+\frac{3}{y}}{\frac{5}{x}-\frac{1}{y}} $$

Step-by-Step Solution

Answer

\( \frac{2y + 3x}{5y - x} \)

Step 1: Identify the Complex Fraction

The given complex fraction is \( \frac{\frac{2}{x}+\frac{3}{y}}{\frac{5}{x}-\frac{1}{y}} \). It consists of a numerator \( \frac{2}{x}+\frac{3}{y} \) and a denominator \( \frac{5}{x}-\frac{1}{y} \), both of which are fractions themselves.

Step 2: Find a Common Denominator

To simplify each part, find the least common denominator (LCD) for the fractions in the numerator and the denominator. For \( \frac{2}{x} \) and \( \frac{3}{y} \), the LCD is \( xy \). For \( \frac{5}{x} \) and \( \frac{1}{y} \), the LCD is also \( xy \).

Step 3: Rewrite Each Fraction

Rewriting each fraction over their common denominator: - Numerator: \( \frac{2}{x} = \frac{2y}{xy} \), \( \frac{3}{y} = \frac{3x}{xy} \) so \( \frac{2}{x} + \frac{3}{y} = \frac{2y + 3x}{xy} \). - Denominator: \( \frac{5}{x} = \frac{5y}{xy} \), \( \frac{1}{y} = \frac{x}{xy} \) so \( \frac{5}{x} - \frac{1}{y} = \frac{5y - x}{xy} \).

Step 4: Simplify the Complex Fraction

Now substitute these into the original complex fraction: \[ \frac{\frac{2}{x} + \frac{3}{y}}{\frac{5}{x} - \frac{1}{y}} = \frac{\frac{2y + 3x}{xy}}{\frac{5y - x}{xy}} \].Since both the numerator and denominator are over the same common denominator \( xy \), they cancel out, leaving: \[ \frac{2y + 3x}{5y - x} \].

Step 5: Check Simplification

Ensure the fraction \( \frac{2y + 3x}{5y - x} \) is fully simplified. Check if there are any common factors to cancel, which there are not. Therefore, this is the simplified form of the original complex fraction.

Key Concepts

Algebra Simplification

Simplifying algebraic expressions often seems challenging, but it's all about taking complex fractions or terms and making them easier to understand and work with. One key aspect is reducing unnecessary complexity in expressions.

For example, algebra simplification involves breaking down complex fractions. In the given exercise, the expression \(\frac{\frac{2}{x} + \frac{3}{y}}{\frac{5}{x} - \frac{1}{y}}\) can be simplified by handling each part separately. By following algebra simplification steps, you're ensuring that any complex terms are expressed in their simplest form.

• Start by identifying all like terms within fractions.
• Apply mathematical operations like addition, subtraction, or simplification of terms.
• Always re-check your results to ensure the simplification is correct and cannot be reduced further.

Breaking the original problem into more manageable parts by simplifying each piece is akin to solving a puzzle one piece at a time!

Least Common Denominator

In fraction operations, especially those involving complex fractions, finding the least common denominator (LCD) is crucial. This is because it allows two or more fractions to be expressed with the same denominator, making them easier to manipulate.

To find the LCD, identify the smallest multiple that each denominator divides into evenly. In the solution provided, the complex fractions \(\frac{2}{x} + \frac{3}{y}\) and \(\frac{5}{x} - \frac{1}{y}\) both have denominators of \( x \) and \( y \).

• The LCD for both sets of fractions is \( xy \), as it's the smallest expression that \( x \) and \( y \) both fit into.
• By converting each fraction to this common denominator, you can combine them through addition or subtraction.
• Doing so simplifies the algebraic manipulation and helps in reducing the expression comprehensively.

Finding the LCD is a pivotal step in tackling fraction operations and reducing the complexity of the problem.

Fraction Operations

Operations with fractions, especially complex fractions, follow a systematic approach. Mastery in fraction operations involves understanding how to add, subtract, multiply, and divide these fractions, along with simplifying them.

First, you need to ensure each fraction is expressed with a common denominator. By doing this, as seen in the original solution, the fractions in the numerator and denominator can easily be rewritten with the same denominators.

• Once rewritten: \(\frac{2}{x} + \frac{3}{y} = \frac{2y + 3x}{xy}\) and \(\frac{5}{x} - \frac{1}{y} = \frac{5y - x}{xy}\)
• This allows for straightforward division by cancelling out the common denominator \( xy \), leading to a simpler fraction \(\frac{2y + 3x}{5y - x}\).
• Always check for any common factors in the final expression that could simplify it further.

Grasping these fraction operations ensures you're equipped for tackling even the most seemingly complex algebra problems with confidence and ease.