Problem 95 - Elementary Algebra

Question

For the following problems, add or subtract the rational expressions. $$ \frac{5 m}{6-m}+\frac{3 m}{m-6} $$

Step-by-Step Solution

Answer

Question: Add the following rational expressions: $$\frac{5m}{6-m} + \frac{3m}{m-6}$$ Answer: $$\frac{2m}{6-m}$$

Step 1: Find the LCD

Since the two denominators are "6 - m" and "m - 6", we notice that they are equal if we multiply the second one by -1. Therefore, the least common denominator (LCD) is 6 - m.

Step 2: Write each fraction with the LCD

The first fraction already has the LCD, so we don't need to change it. For the second fraction, we need to multiply both the numerator and the denominator by -1 in order to obtain the LCD. This results in the following expression: $$ \frac{5m}{6-m} - \frac{3m}{6-m} $$

Step 3: Add the fractions

Now we can combine the two fractions since they have the same denominator (6-m): $$ \frac{5m - 3m}{6-m} = \frac{2m}{6-m} $$ So, the final solution is: $$ \frac{5m}{6-m} + \frac{3m}{m-6} = \frac{2m}{6-m} $$

Key Concepts

Understanding the Least Common Denominator

When dealing with rational expressions, finding the least common denominator (LCD) is crucial for simplifying fraction addition and subtraction. The LCD is the smallest expression that can be a common denominator for all fractions involved.
To determine the LCD, you look at the denominators of each fraction. In this exercise’s case, the denominators are \(6 - m\) and \(m - 6\). Although these look different, they are surprisingly similar.
Notice that \(m - 6\) can be transformed into \(6 - m\) by multiplying it by \(-1\). Hence, the LCD is \(6 - m\). This allows you to manipulate the fractions to have the same denominator, facilitating the next steps in fraction operations.

Fraction Addition and Subtraction with Rational Expressions

Adding and subtracting fractions requires a common denominator, especially when dealing with rational expressions. Once a common denominator is found—like in the given exercise—the expression becomes much simpler to handle.
Here’s how it works:

• Adjust one or both fractions so they have the same denominator. In this exercise, \(\frac{5m}{6-m}\) already had the common denominator, while \(\frac{3m}{m-6}\) needed adjustment by multiplying the numerator and denominator by \(-1\).
• Once the denominators match, you combine the fractions by adding or subtracting their numerators, keeping the denominator constant. Here, it changes from separate fractions to \(\frac{5m - 3m}{6-m}\).
• Simplify the resulting fraction if possible. In this solution, the simplified answer becomes \(\frac{2m}{6-m}\).

This process illustrates the importance of the LCD and shows how it leads to easier addition or subtraction of algebraic expressions.

Dealing with Algebraic Expressions

Algebraic expressions are expressions that involve variables and constants combined using mathematical operations. In the context of rational expressions like these fractions, understanding how to manipulate variables is essential.
The algebraic expression in the numerator can often be simplified by combining like terms or performing arithmetic operations. For instance, in the exercise, after rewriting the problem as fractions with a common denominator, the numerators \(5m\) and \(3m\) are simplified to \(2m\) by subtraction.
This requires a good grasp of basic algebraic principles such as distributing, multiplying, and combining similar terms—skills that are vital in further algebraic problem-solving processes. Being comfortable with these concepts is key when dealing with more complex expressions. By carefully following these steps, solving rational expression problems becomes less daunting and more understandable.