Problem 23 - Elementary Algebra

Question

Add or subtract as indicated. Be sure to express your answers in simplest forn. (Objective 1) $$\frac{5 x-2}{7 x}-\frac{8 x+3}{7 x}$$

Step-by-Step Solution

Answer

\( \frac{-3x - 5}{7x} \)

Step 1: Identify the Expression and Determine the Operation

The given expression is \( \frac{5x-2}{7x} - \frac{8x+3}{7x} \). You are required to subtract one fraction from the other.

Step 2: Align the Denominators

Both terms have the same denominator, which is \( 7x \). Therefore, you can proceed directly with the subtraction of the numerators without needing to find a common denominator.

Step 3: Subtract the Numerators

Subtract the entire numerator of the second fraction from the numerator of the first: \((5x - 2) - (8x + 3)\). Ensure to distribute the negative sign across the second numerator.

Step 4: Simplify the Resulting Numerator

Perform the subtraction: \[ (5x - 2) - (8x + 3) = 5x - 2 - 8x - 3 = -3x - 5 \].

Step 5: Form the Simplified Fraction

Combine the simplified numerator with the common denominator: \( \frac{-3x - 5}{7x} \).

Step 6: Verify the Simplified Form

The fraction is simplified since there are no common factors other than 1 in the numerator and the denominator. Thus, \( \frac{-3x - 5}{7x} \) is the simplest form.

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential skill in algebra, and it involves reducing fractions to their simplest form. When we simplify a fraction, we aim to make the numerator (the top number) and the denominator (the bottom number) as small as possible while still retaining the same value.To simplify a fraction, follow these steps:

• Find the greatest common divisor (GCD) of both the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.
• The new fraction is the simplest form, as long as the numerator and denominator share no common factors other than 1.

Simplifying fractions can make calculations easier and helps in understanding the expression better. In the solution provided, the fraction \( \frac{-3x - 5}{7x} \) is already in its simplest form as there are no common factors between \(-3x - 5\) and \(7x\). This confirms that the answer is correctly simplified.

Subtraction of Fractions

Subtracting fractions involves a few straightforward steps, but it can sometimes be tricky if the denominators are not already the same. The subtraction of fractions includes several important concepts:

• Common Denominator: Ensure both fractions have the same denominator before subtracting the numerators.
• Subtract Numerators: After aligning the denominators, subtract the numerator of one fraction from the numerator of the other.
• Combine and Simplify: Combine the result over the common denominator, and then simplify the resulting fraction if possible.

In our exercise, the denominators were conveniently the same (i.e., \(7x\)), allowing us to skip directly to subtracting the numerators: \((5x - 2) - (8x + 3)\). It is crucial to pay attention to distributing the negative sign correctly across the second fraction's numerator to avoid errors. Once you perform the subtraction, you'll get \(-3x - 5\), forming the simplified fraction \( \frac{-3x - 5}{7x} \).

Step-by-Step Math Solutions

Step-by-step math solutions are a powerful tool for understanding complex algebraic problems. They allow students to follow the logic and methodology behind each step in a problem-solution process. Here's why it matters:

• Logical Progression: Breaking down the solution into steps helps reveal the logical progression of solving the problem.
• Clarity and Understanding: Each step ensures clarity, making it easier to pinpoint where mistakes might occur or where additional understanding is needed.
• Learning Reinforcement: Going through each step reinforces learning and helps embed the process in memory.

By using a step-by-step approach for the original exercise, students can see how to handle each part of the subtraction of algebraic fractions effectively. They learn how to keep track of numerators and denominators, manage signs during subtraction, and ultimately how to simplify their final answer correctly. This method demystifies complex problems, making math more accessible and less intimidating.