Problem 53 - Algebra for College Students

Question

For Problems 51-58, simplify each rational expression. You will need to use factoring by grouping. \(\frac{a x-3 x+2 a y-6 y}{2 a x-6 x+a y-3 y}\)

Step-by-Step Solution

Answer

The simplified expression is \(\frac{x + 2y}{2x + y}\), valid for \(a \neq 3\).

Step 1: Analyze the Expression

We are given the rational expression \(\frac{a x-3 x+2 a y-6 y}{2 a x-6 x+a y-3 y}\). The goal is to simplify it by factoring each part of the fraction.

Step 2: Factor the Numerator by Grouping

The numerator is \(a x - 3 x + 2 a y - 6 y\). We first group the terms: \((a x - 3 x) + (2 a y - 6 y)\). For the first group \((a x - 3 x)\), factor out \(x\), resulting in \(x(a - 3)\). For the second group \((2 a y - 6 y)\), factor out \(2y\), resulting in \(2 y(a - 3)\). Thus, the numerator becomes \((a - 3)(x + 2y)\).

Step 3: Factor the Denominator by Grouping

The denominator is \(2 a x - 6 x + a y - 3 y\). Group the terms as \((2 a x - 6 x) + (a y - 3 y)\). For the first group \((2 a x - 6 x)\), factor out \(2x\), resulting in \(2 x(a - 3)\). For the second group \((a y - 3 y)\), factor out \(y\), resulting in \(y(a - 3)\). Thus, the denominator becomes \((a - 3)(2x + y)\).

Step 4: Simplify the Rational Expression

Now, rewrite the rational expression with the factored numerator and denominator: \(\frac{(a - 3)(x + 2y)}{(a - 3)(2x + y)}\). Since \((a - 3)\) is a common factor in both the numerator and the denominator, they can be canceled out, leaving \(\frac{x + 2y}{2x + y}\).

Step 5: Check for Division by Zero

The simplification is only valid if \(a - 3 eq 0\). Therefore, the solution holds as long as \(a eq 3\).

Key Concepts

Simplifying Rational Expressions

Rational expressions are similar to fractions but involve polynomials instead of simple integers in the numerator and denominator. Simplifying these expressions is crucial to making complex algebraic problems more manageable. It involves reducing the expression to its lowest possible terms by eliminating any common factors that exist between the numerator and the denominator.
To simplify a rational expression:

• Factor both the numerator and the denominator completely if possible.
• Identify any common factors that appear in both the numerator and the denominator.
• Cancel out these common factors.

The process isn't fully complete until you ensure that there are no constraints violated, such as division by zero. Therefore, after simplifying, revisit the original expression and check what variable values could result in a denominator of zero to avoid undefined expressions. For example, when simplifying \( \frac{(a - 3)(x + 2y)}{(a - 3)(2x + y)} \), we cancel \( a - 3 \) because it's common to both the numerator and denominator, but must ensure \( a eq 3 \). This keeps the expression valid by avoiding zero in the denominator.

Factoring Techniques

Factoring is the process of breaking down a complex expression into simpler, multiplied expressions or factors. It's a vital technique used when simplifying algebraic expressions. One of the most common techniques is factoring by grouping, which is particularly useful for simplifying expressions like the one in our original exercise.
Factoring by grouping involves:

• Organizing the expression in a way that pairs certain terms together.
• Within each pair, factor out a common factor.
• Rewrite the expression as a product of these factors.

Consider the numerator from the step-by-step solution: \( ax - 3x + 2ay - 6y \). By grouping as \( (ax - 3x) + (2ay - 6y) \) and factoring each section, \( x(a - 3) \) and \( 2y(a - 3) \), we consolidate it as \( (a - 3)(x + 2y) \). Similarly, the denominator can be managed in the same manner. These techniques are foundations that allow simplification of even more elaborate algebraic expressions.

Algebraic Expressions

Algebraic expressions consist of variables, constants, and operations such as addition, subtraction, multiplication, and division. They form the basis for algebra, enabling representation and solving of real-world problems in a symbolic manner. Understanding how to manipulate these expressions, including simplifying or factoring them, is essential in algebra.
Characteristics of algebraic expressions include:

• Terms: In our expression, each separate part like \( ax \) or \( -3x \) is a term, which can be a single number, a variable, or a product of both.
• Coefficients: These are the numerical parts of terms, such as 3 in \( 3x \).
• Variables: Symbols like \( x \) or \( y \) that represent unknown values.

Algebraic expressions can range from very simple to complex, requiring mastering different levels of manipulation including addition, subtraction, or more advanced methods like factoring. These skills are fundamental not just in mathematical exercises but also serve in advanced topics like calculus and real-world problem-solving.