Problem 14 - Algebra A Combined Function

Question

$$ \frac{3 x-1}{x^{2}+5 x-6}-\frac{2 x-7}{x^{2}+5 x-6} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{x + 6}{x^2 + 5x - 6} \).

Step 1: Identify Common Denominator

Notice that both fractions share the same common denominator, which is \(x^2 + 5x - 6\). This means we can combine the numerators directly.

Step 2: Combine the Numerators

Since the denominators are the same, we combine the numerators: \( (3x - 1) - (2x - 7) \). Be careful with the negative sign, which applies to both terms in the second numerator.

Step 3: Simplify the Numerator

Simplify the new numerator by distributing the negative sign and combining like terms: \(3x - 1 - 2x + 7 = (3x - 2x) + (-1 + 7) = x + 6 \).

Step 4: Write the Combined Fraction

Combine the result into a single fraction: \( \frac{x + 6}{x^2 + 5x - 6} \). This is the simplified form of the expression.

Key Concepts

Common Denominator

A common denominator is a crucial concept when dealing with rational expressions. In the original exercise, both rational expressions share the same denominator, which is \(x^2 + 5x - 6\). This simplifies the process of subtraction between the rational numbers because we do not need to manipulate the denominator to be the same for both terms. To understand this concept better, think of a denominator as the part of a fraction that represents how "whole" that part is divided.

- When two fractions have a common denominator, they can be directly combined by performing operations on the numerators.- Without a common denominator, we'd need to find one, usually by taking the least common multiple of the current denominators.- In our case, the denominator \(x^2 + 5x - 6\) is already the same for both fractions, allowing a straightforward operation on the numerators.

A shared denominator is not only about simplifying calculations but also provides a unified ground for expressing differences or sums within rational expressions.

Numerator and Denominator

The terms "numerator" and "denominator" are essential for understanding fractions and rational expressions. In a fraction, the numerator is the top part, whereas the denominator is the bottom part. The original exercise demonstrates these principles in the expression \( \frac{3x-1}{x^2 + 5x - 6} \) minus \( \frac{2x-7}{x^2 + 5x - 6} \). Here, \(3x - 1\) and \(2x - 7\) are numerators, while \(x^2 + 5x - 6\) remains as the denominator.

- Numerators tell us "how many parts" we have.- Denominators tell us "what size" the parts are.

When both fractions have the same denominator, operations involve the numerators only. In our given expression, this operation is subtraction where the numerator \(3x - 1\) has the other numerator \(2x - 7\) subtracted from it. Ensuring that these parts are clearly understood helps in performing accurate mathematical operations.

Simplifying Expressions

Simplifying expressions is a method of rewriting an expression in its most basic form. It's about making expressions easier to work with while preserving their value. The goal is to combine like terms and reduce expressions to find simpler equivalent expressions.

In the original problem, we see simplification in action in the step where numerators are combined: \((3x - 1) - (2x - 7)\). This step involves distributing the negative sign across \(2x - 7\) resulting in \(3x - 1 - 2x + 7\).

- Combine like terms: \((3x - 2x) + (-1 + 7)\). - The result simplifies to \(x + 6\).

This new numerator \(x + 6\) is then over the shared denominator \(x^2 + 5x - 6\), giving us the simplified form \( \frac{x + 6}{x^2 + 5x - 6} \). Simplifying expressions in mathematics helps in making calculations more manageable and in gaining insights into the nature of the expressions.