Problem 65 - Elementary Algebra

Question

Simplify each fraction. You will need to use factoring by grouping. $$\frac{x y-6 x+y-6}{x y-6 x+5 y-30}$$

Step-by-Step Solution

Answer

The fraction simplifies to \(\frac{x + 1}{x + 5}\).

Step 1: Identify Common Factors in the Numerator

The numerator is \(xy - 6x + y - 6\). Group the terms to find a common factor: \((xy - 6x) + (y - 6)\). Factor out the common factors in each group: \(x(y - 6) + 1(y - 6)\). Notice \(y - 6\) is a common factor, so factor it out: \((x + 1)(y - 6)\).

Step 2: Identify Common Factors in the Denominator

The denominator is \(xy - 6x + 5y - 30\). Group the terms to find a common factor: \((xy - 6x) + (5y - 30)\). Factor out the common factors in each group: \(x(y - 6) + 5(y - 6)\). Notice \(y - 6\) is a common factor, so factor it out: \((x + 5)(y - 6)\).

Step 3: Simplify the Fraction

The fraction now is \(\frac{(x + 1)(y - 6)}{(x + 5)(y - 6)}\). Cancel out the common factor \((y - 6)\) in the numerator and the denominator. The simplified form of the fraction is \(\frac{x + 1}{x + 5}\).

Key Concepts

Simplifying Fractions

Simplifying fractions is a critical skill when working with algebraic expressions. It involves reducing a fraction to its simplest form, making it easier to work with. The key to simplifying is to identify and eliminate any common factors between the numerator and the denominator.

• Identify: Look for common terms or factors in both the numerator and the denominator.
• Factor: Often, this involves some form of factoring, such as grouping, to find common terms.
• Cancel: Once common terms are identified, they can be canceled out. This step reduces the complexity of the fraction.

In our exercise, we initially have complex algebraic expressions in both the numerator and the denominator. We restructure them by using factoring by grouping, which helps in finding a common term that can be canceled out, simplifying the equation significantly.

Common Factors

Common factors are terms or expressions that are present in both parts of a fraction. They play a pivotal role in simplifying fractions, as identifying these factors allows us to reduce a fraction to its simplest state.

• Find: Focus on terms or expressions that multiply to give both quantities in the fraction.
• Group: Sometimes grouping terms together can reveal these common factors more clearly.
• Factor Out: Once identified, factor out these common components, simplifying the equation.

In the original problem, we see the terms (xy - 6x) and (y - 6) in the numerator, while the denominator has (xy - 6x) and (5y - 30). Upon factoring, both grouping expressions revealed a common factor, allowing us to simplify the entire fraction. This approach showcases the importance of carefully analyzing each component of the algebraic expressions involved.

Algebraic Expressions

Algebraic expressions consist of variables and constants combined using arithmetic operations. In exercises involving algebraic fractions, understanding the nature of these expressions is crucial for successful simplification.

• Components: They generally include terms like coefficients, variables, and exponents.
• Operations: Factors can include addition, subtraction, multiplication, or division.
• Combination: Complex expressions often need manipulation, such as factoring, to simplify.

In the provided problem, we break down the expressions like \(xy - 6x + y - 6\) into smaller parts to make it easier to handle. Recognizing the expressions and learning how to manipulate algebraic terms are fundamental to carrying out operations like factoring by grouping. This knowledge is essential for ensuring accuracy and efficiency in algebraic computations.