Problem 64 - Intermediate Algebra

Question

Perform the operations and simplify the result when possible. See Example \(8 .\) $$\frac{4}{a+4}-\frac{2 a}{a+4}+\frac{3 a}{a+4}$$

Step-by-Step Solution

Answer

The simplified result is \( \frac{4+a}{a+4} \).

Step 1: Identify the Common Denominator

The fractions \( \frac{4}{a+4} \), \( -\frac{2a}{a+4} \), and \( \frac{3a}{a+4} \) already have a common denominator, which is \( a+4 \). This allows us to combine these fractions easily.

Step 2: Combine the Numerators

Since the denominators are the same, subtract and add the numerators: \( 4 - 2a + 3a \). Simplify this expression by combining like terms in the numerator, resulting in \( 4 + a \).

Step 3: Write the Combined Fraction

The expression now simplifies to one fraction: \( \frac{4 + a}{a+4} \).

Step 4: Simplify the Expression (if possible)

Check if the fraction \( \frac{4+a}{a+4} \) can be further simplified. In this case, it cannot be simplified further as there are no common factors between the numerator \( 4+a \) and the denominator \( a+4 \).

Key Concepts

Algebraic Fractions

Algebraic fractions are similar to regular fractions, but they introduce variables into the mix. Just like with numerical fractions, algebraic fractions consist of a numerator and a denominator, each of which can be an expression. In this exercise, each fraction has a common denominator of \( a+4 \). This simplifies the process of combining them. The main idea is to think of algebraic fractions as just numbers, but with letters involved, representing unknown or variable quantities. This approach helps in performing operations such as addition and subtraction.

Combining Like Terms

Combining like terms involves simplifying algebraic expressions by merging terms that contain the same variables raised to the same power. In the given problem, the expression "\( 4 - 2a + 3a \)" appears in the numerator. Each term in this expression is either a constant or involves the variable \( a \).

To simplify, perform the operation for the variable terms: \( -2a + 3a = a \). This is because you are simply adding and subtracting the coefficients of the like terms. The constant term, \( 4 \), remains unchanged, resulting in a simplified numerator of \( 4 + a \). This step is crucial as it prepares the expression for further simplification or operations.

Common Denominator

In order to add or subtract algebraic fractions, they must have a common denominator. This commonality allows us to directly manipulate the numerators while keeping the denominator the same. The fractions \( \frac{4}{a+4}, \frac{-2a}{a+4}, \text{and} \frac{3a}{a+4} \) share the denominator \( a+4 \), making the process straightforward.

Once the denominators match, you proceed by focusing on the numerators. By doing so, you avoid complications that arise from adding or subtracting fractions with different bases. The outcome is a single fraction, where the combined numerators display the simplified outcome directly linked to the original expression. Keeping the denominator unchanged simplifies not just the arithmetic but also the visual clarity of the problem.