Problem 63 - Algebra A Combined Function

Question

\(\frac{8 x+7}{3 x+5}-\frac{2 x-3}{3 x+5}\)

Step-by-Step Solution

Answer

The expression simplifies to 2.

Step 1: Identify the Common Denominator

In the expression \( \frac{8x + 7}{3x + 5} - \frac{2x - 3}{3x + 5} \), both fractions have the same denominator, which is \( 3x + 5 \). This means we can combine the numerators directly.

Step 2: Combine the Numerators

Since the denominators are the same, subtract the numerators: \( (8x + 7) - (2x - 3) \). Note that when subtracting, each term in the second numerator changes its sign.

Step 3: Simplify the Combined Numerator

Distribute the negative sign in the second term: \( 8x + 7 - 2x + 3 \). Now combine like terms. The \( x \) terms \( 8x - 2x \) give \( 6x \) and the constant terms \( 7 + 3 \) give \( 10 \). This yields \( 6x + 10 \).

Step 4: Express the Simplified Fraction

Place the simplified numerator over the common denominator: \( \frac{6x + 10}{3x + 5} \).

Step 5: Verify and Simplify Further if Possible

Check if the expression \( 6x + 10 \) can be factored. It can be factored as \( 2(3x + 5) \). The fraction \( \frac{6x + 10}{3x + 5} \) then simplifies to \( \frac{2(3x + 5)}{3x + 5} = 2 \), since \( 3x + 5 \) in the numerator and the denominator cancel each other out.

Key Concepts

Common Denominator

Understanding the concept of a common denominator is key when working with rational expressions involving fractions. The denominator is the bottom part of a fraction, which tells you how many parts make up a whole. When adding or subtracting fractions, it is essential to have a common denominator because it allows you to directly combine the numerators. In the exercise, both fractions share the denominator \(3x + 5\). Since they are the same, we don’t have to find a new common denominator. This simplifies our task because it means we can focus directly on what is happening in the numerator. To do this:- Identify if the fractions in your expression already have the same denominator.- If they do, you can proceed to combine the numerators directly.- If they don’t, you’d need to find a common denominator, often the least common multiple of the denominators.

Simplifying Fractions

Once you have a shared denominator, the next step is to simplify the resulting fraction. Simplifying fractions involves condensing the expression into its most reduced form. This can make calculations easier and also helps in understanding the behavior of the expression. In the problem, after combining the numerators to form \(6x + 10\), the next step is checking whether this can be simplified. Simplification often occurs by combining like terms, which means gathering and summing terms that have the same variable part.Steps to simplify:- Combine like terms by adding or subtracting coefficients of terms that have the same variable component.- Once combined, place back over the common denominator, as shown in the original step where \(6x + 10\) is placed over \(3x + 5\).- Check if further reduction is possible, often through factoring, which is the next step.

Factoring

Factoring is when you rewrite a number or an expression as a product of its factors. This is an essential skill in algebra because it allows you to simplify expressions and solve equations. In the exercise, we see that the numerator \(6x + 10\) can be factored. Factoring involves finding common factors in the expression. Here, \(6x + 10\) can be expressed as \(2(3x + 5)\) because both 6 and 10 are divisible by 2. This factorization is crucial as it reveals a common factor in the numerator and the denominator that can be cancelled. The steps to factoring include:- Identifying common factors among the terms of a polynomial.- Extract the greatest common factor, in this case, 2 from both \(6x\) and \(10\).- Rewrite the expression using the factor, leading to \(2(3x + 5)\).After factoring, if the same expression appears in both the numerator and the denominator, they can be cancelled out, simplifying the fraction further, which results in the simplicity of the expression to just \(2\) in this case.