Problem 37 - Elementary and Intermediate Algebra

Question

Subtract and simplify the result, if possible. \(\frac{6 x-5}{3 x y}-\frac{3 x-5}{3 x y}\)

Step-by-Step Solution

Answer

\(\frac{1}{y}\)

Step 1: Identify the Problem

The problem is to subtract two rational expressions with the same denominator:\[\frac{6x-5}{3xy} - \frac{3x-5}{3xy}\]

Step 2: Subtract the Numerators

Since both fractions have the same denominator \(3xy\), we can subtract the numerators directly:\[(6x-5) - (3x-5)\]Make sure to distribute the negative sign through the second numerator.

Step 3: Simplify the Numerator

Perform the subtraction in the numerator:\[6x - 5 - 3x + 5 = (6x - 3x) + (-5 + 5)\]This simplifies to:\[3x + 0 = 3x\]

Step 4: Write the Simplified Fraction

Place the simplified numerator over the common denominator:\[\frac{3x}{3xy}\]

Step 5: Simplify the Fraction

We can simplify this fraction by canceling the common factor of \(3x\) from the numerator and the denominator:\[\frac{3x}{3xy} = \frac{1}{y}\]

Step 6: Confirm the Simplification

The final simplified expression is \(\frac{1}{y}\). There are no further common factors to simplify.

Key Concepts

Rational Expressions

Rational expressions are similar to fractions but with polynomials in the numerator and denominator. Just like fractions, they can be added, subtracted, multiplied, or divided, providing us with the flexibility to manipulate algebraic expressions.
Understanding rational expressions involves recognizing that the denominator cannot be zero, as division by zero is undefined. This is crucial when simplifying or solving equations involving rational expressions, ensuring that any proposed solution does not result in an undefined expression.
Also, rational expressions create opportunities to analyze relationships within algebraic structures. They offer insight into how variables behave when combined in multiple-term equations.

Subtracting Fractions

Subtracting fractions, whether simple or involving algebraic terms, follows a similar rule: only subtract the numerators when the denominators are the same. For rational expressions, this concept remains crucial.
If you subtract one fraction from another, it’s essential to ensure they share a common denominator—this is the only way to directly subtract across the numerators. For example, in our exercise:

• The two expressions have a common denominator of \(3xy\).
• Thus, we subtract the numerators as \(6x-5) - (3x-5)\).

Carefully distribute any negative signs through the parentheses to maintain accuracy as the numerators are combined.

Simplifying Fractions

Simplifying fractions involves reducing them to their most basic form. This means removing common factors from the numerator and denominator.
After performing subtraction in the previous step, we are left with \(\frac{3x}{3xy}\). The simplification process involves:

• Identifying common factors in both numerator (\(3x\)) and denominator (\(3xy\)).
• Canceling out these common factors to achieve the simplest form, resulting in \(\frac{1}{y}\).

This simplicity is useful for further calculations, offering clearer solutions that are less prone to errors in more complex expressions.

Like Denominators

Like denominators are key when working with fractions or rational expressions. They allow us to directly add or subtract the numerators without further adjustments.
Achieving this condition enables smoother mathematical operations, avoiding complex procedures to align expressions for arithmetic processes. In complex expressions:

• Equating denominators might mean multiplying terms to harmonize them when they initially differ.
• Once aligned, operations like subtraction or addition can proceed straightforwardly.

The concept of like denominators simplifies what might otherwise be a daunting task, promoting ease and accuracy in algebraic manipulations.