Problem 28 - Algebra A Combined Function

Question

Perform each indicated operation. Simplify if possible. \(\frac{7 x}{x-3}-\frac{4 x+9}{x-3}\)

Step-by-Step Solution

Answer

The simplified result is 3.

Step 1: Identify the Common Denominator

Observe that both fractions have the same denominator \(x - 3\). This means you can combine the fractions directly.

Step 2: Combine the Numerators

Since the denominators are the same, combine the numerators: \(7x\) and \(-(4x + 9)\). This gives \(7x - (4x + 9)\).

Step 3: Simplify the Numerator

Distribute the negative sign in the second term and then combine like terms: \(7x - 4x - 9 = 3x - 9\).

Step 4: Write the Resulting Fraction

The simplified form of the combined fraction is \(\frac{3x - 9}{x-3}\).

Step 5: Factor and Simplify Further (if possible)

Factor the numerator: \(3(x-3)\). The expression becomes \(\frac{3(x-3)}{x-3}\). Cancel the \(x-3\) terms, giving the final result: \(3\).

Key Concepts

Fractions in Algebraic Expressions

Working with fractions within algebraic expressions can seem daunting at first, yet understanding their structure can significantly ease the process. Fractions consist of a numerator and a denominator. In the given exercise, these fractions had common denominators, which is a convenient advantage for simplification. When fractions share the same denominator, you can directly combine their numerators:

• The numerator from the first fraction minus the numerator of the second.
• This simplifies the operation to simple arithmetic on the numerators alone.

Keeping track of variables and constant terms while managing signs correctly is key. Using this method avoids errors and leads to a smoother simplification process. The shared denominator effectively acts as a common baseline for the entire equation, making it a systematic approach in algebraic fraction manipulations.
As you encounter fractions with different denominators, the process will involve finding a common denominator to convert them to a similar form as the first step, ensuring your path to simplification is clear.

Simplification of Algebraic Expressions

Simplification is a crucial operation in algebra that enhances readability and clarity in expressions. It involves combining like terms, reducing complex forms, and making the expression as straightforward as possible. In our step-by-step solution:

• We combined like terms: Terms with the same variable raised to the same power.
• This included merging terms by arithmetic operations like addition and subtraction.

Executing the simplification requires careful attention to details like distribution of negative signs, which can easily alter the outcome. For instance, when we simplified the numerator, \(7x - (4x + 9)\), distributing the negative reversed the signs for both \(4x\) and \(9\).
Simplifying algebraic expressions involves performing arithmetic cleanly while keeping variable powers consistent. Achieving a simplified form often makes further solutions, such as solving equations or analyzing graphs, more evident and approachable.

Factoring in Algebra

Factoring is a method used in algebra to simplify expressions further or solve equations. In the exercise, the final simplification step involved factoring the numerator, \(3x - 9\), which was expressed as \(3(x - 3)\). This step involved:

• Identifying common factors in the terms of the numerator, in this case, \(3\).
• Expressing the numerator as a product of that common factor and a simplified remaining term.

Factoring is an effective tool as it can sometimes reveal common terms in the numerator and denominator that can be canceled. This not only simplifies the expression further but also helps in solving equations where these expressions are part of larger solutions. By removing these common factors, we arrive at a simpler form, which, in the given exercise, was reduced to the constant \(3\). Factoring hence becomes a transformative process in algebra, facilitating easier manipulation and deeper insights into the properties of expressions and equations.