Problem 45 - Algebra A Combined Function

Question

Perform each indicated operation. Simplify if possible. \(\frac{3 a}{2 a+6}-\frac{a-1}{a+3}\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{a + 2}{2(a+3)} \).

Step 1: Identify common denominator

The given expressions are \( \frac{3a}{2a+6} \) and \( \frac{a-1}{a+3} \). First, notice that the denominator \( 2a+6 \) can be factored as \( 2(a+3) \). So, the denominators are \( 2(a+3) \) and \( a+3 \). The common denominator for both fractions is \( 2(a+3) \).

Step 2: Rewrite fractions with common denominator

Rewrite each fraction with the common denominator \( 2(a+3) \): - The first fraction is already with this common denominator.- The second fraction \( \frac{a-1}{a+3} \) needs to be rewritten. Multiply both the numerator and the denominator by 2 to get \( \frac{2(a-1)}{2(a+3)} \).

Step 3: Subtract the fractions

Now subtract the fractions: \( \frac{3a}{2(a+3)} - \frac{2(a-1)}{2(a+3)} \). Since they have a common denominator, subtract their numerators only: \[\frac{3a - 2(a-1)}{2(a+3)}\]

Step 4: Simplify the expression

Now simplify the expression:\[3a - 2(a-1) = 3a - 2a + 2 = a + 2\]So, the simplified numerator is \( a+2 \).

Step 5: Write the final simplified expression

The simplified form of the original expression is: \[\frac{a + 2}{2(a+3)}\]

Step 6: Examine further simplification

Check if \( \frac{a + 2}{2(a+3)} \) can be simplified further. Since there are no common factors, this is the simplest form.

Key Concepts

Simplifying Expressions

Simplifying expressions is an essential skill in algebra. It involves rewriting complex expressions into more manageable forms without changing their value. When simplifying an algebraic expression, the goal is to reduce it to as few terms as possible.

To simplify an expression like \( \frac{3a}{2a+6} - \frac{a-1}{a+3} \), we first identify if the expression can be reduced, rearranged, or combined. By consolidating terms and reducing fractions, the expression becomes clearer and easier to work with.

• Firstly, examine the structure: Look at the numerators and denominators for common factors.
• Next, apply arithmetic operations: Use addition, subtraction, multiplication, or division to combine or reduce terms.
• Finally, rewrite the expression: The result should be in a simplified form, free from needless complexity.

Simplifying helps make subsequent steps like solving or further manipulation easier. Always ensure that you compare initial and final expressions to confirm correctness.

Common Denominator

The concept of a common denominator is crucial when handling fractions, especially when adding or subtracting them. A common denominator is a shared multiple of the denominators in two or more fractions.

To successfully subtract \( \frac{3a}{2a+6} - \frac{a-1}{a+3} \), we need a common denominator. This involves:

• Factoring each denominator: Break down each denominator to see its factors. For instance, \( 2a+6 \) factors into \( 2(a+3) \).
• Identifying the least common multiple: In this case, it is \( 2(a+3) \) because it encompasses both denominators' factors.
• Adjusting each fraction: Rewrite fractions so they have the common denominator by multiplying the numerator and denominator appropriately.

Working with a common denominator allows you to combine or compare fractions directly. It simplifies the calculation by ensuring you're working with like terms.

Factoring

Factoring is a method used in algebra to simplify expressions by breaking them down into their constituent components. It involves expressing a number or expression as a product of its factors, making it easier to handle.

For instance, in the expression \( \frac{3a}{2a+6} \), the denominator can be factored into \( 2(a+3) \), which reveals a common factor: \( a+3 \). Here’s how factoring can help:

• Factor recognition: Quickly spot patterns that can be broken down (e.g., common numerical factors, differences of squares, or expression sum/difference patterns).
• Facilitates simplification: Once factors are identified, any common factors between numerator and denominator can be canceled out, simplifying the expression further.
• Precursor to solving: Factoring sets expressions up for other operations, such as finding roots or further simplification.

Factoring is not just a step towards simplification; it's a foundational technique that provides clarity and further manipulation of algebraic tasks.