Problem 7 - Algebra 2

Question

Simplify each expression. $$ \frac{5}{3 m}-\frac{2}{7 m}-\frac{1}{2 m} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{37}{42m}\).

Step 1: Identify the Least Common Denominator (LCD)

Identify the least common denominator for the fractions involved. The denominators in this expression are \(3m\), \(7m\), and \(2m\). The least common multiple of \(3\), \(7\), and \(2\) is \(42\). Therefore, the least common denominator (LCD) is \(42m\).

Step 2: Convert Each Fraction to Have the LCD

Convert each fraction so that each has the same denominator, \(42m\). For \(\frac{5}{3m}\), multiply both the numerator and the denominator by \(14\) to get \(\frac{70}{42m}\).For \(\frac{2}{7m}\), multiply both the numerator and the denominator by \(6\) to get \(\frac{12}{42m}\).For \(\frac{1}{2m}\), multiply both the numerator and the denominator by \(21\) to get \(\frac{21}{42m}\).

Step 3: Subtract Fractions

Subtract each fraction now that they have a common denominator:\[\frac{70}{42m} - \frac{12}{42m} - \frac{21}{42m} = \frac{70 - 12 - 21}{42m} = \frac{37}{42m}.\]

Step 4: Write the Final Simplified Expression

The final expression after performing the subtraction is already simplified as \(\frac{37}{42m}\). There are no further factors in the numerator and denominator that can be simplified.

Key Concepts

Least Common Denominator

In algebra, when dealing with fractions, finding the least common denominator (LCD) is a crucial step. The LCD helps in adding or subtracting fractions by transforming them into equivalent fractions with the same denominator. For the exercise with the terms \( \frac{5}{3m} \), \( \frac{2}{7m} \), and \( \frac{1}{2m} \), the denominators are \( 3m \), \( 7m \), and \( 2m \). To find the LCD:

• First, identify the least common multiple (LCM) of the numerical parts, which are 3, 7, and 2. The LCM is the smallest number that can be divided evenly by each of these numbers. Here, it is 42.
• Since each fraction also includes the variable \( m \), it's part of the LCD as well. So, the full expression for the LCD becomes \( 42m \).

This LCD will "unify" the denominators, making it easier to combine or subtract the fractions. An LCD ensures you are comparing fractions on a common scale.

Fractions in Algebra

Working with fractions in algebra is similar to working with any other kind of fractions, but there's an added layer of complexity due to variables. When simplifying expressions like \( \frac{5}{3m} - \frac{2}{7m} - \frac{1}{2m} \), it is important to manage both numerical and variable parts of the fraction.

• Each fraction contains a numerator and a denominator, like a standard fraction.
• To manipulate them, you have to perform operations across the entire fraction (e.g., multiplying both numerator and denominator).
• The goal is to "standardize" the expression by transforming all fractions to have a common denominator, such as \( 42m \) in our case.

This process requires careful attention to how variable terms are handled, ensuring mathematical relations are preserved. Understanding these principles is key for algebraic proficiency.

Algebraic Subtraction

Algebraic subtraction involves combining similar terms across expressions, but it requires them to be in a comparable form, often with a common denominator. Once the fractions in an expression share the same denominator, subtracting them becomes straightforward. Using the equivalent fractions from our exercise:

• Write fractions with the LCD: \( \frac{70}{42m} \), \( \frac{12}{42m} \), \( \frac{21}{42m} \).
• Now subtract directly: \( \frac{70 - 12 - 21}{42m} \).
• The result is \( \frac{37}{42m} \).

Each step respects the arithmetic rules for operations on fractions, ensuring that only the numerators are subtracted while the common denominator remains unchanged. Double-check that no further simplification is possible to confirm the fraction is in its simplest form, as demonstrated by \( \frac{37}{42m} \).