Problem 20 - Elementary Algebra

Question

Add or subtract as indicated. Be sure to express your answers in simplest forn. (Objective 1) $$\frac{2 n-5}{10}+\frac{6 n-1}{10}$$

Step-by-Step Solution

Answer

\( \frac{4n - 3}{5} \)

Step 1: Identify Common Denominators

Both fractions, \( \frac{2n-5}{10} \) and \( \frac{6n-1}{10} \), have the same denominator, which is 10. This allows us to add the numerators directly.

Step 2: Add the Numerators

Since the denominators are the same, we simply add the numerators: \((2n - 5) + (6n - 1)\). This simplifies to \(2n + 6n - 5 - 1\), resulting in \(8n - 6\).

Step 3: Express the Resulting Fraction

Combine the result from adding the numerators over the common denominator: \( \frac{8n - 6}{10} \).

Step 4: Simplify the Fraction

Look for any common factor in the numerator \(8n - 6\) and the denominator 10. Both 8, 6, and 10 are divisible by 2. Factoring 2 out of the numerator, you get \( 2(4n - 3) \). Simplify \( \frac{2(4n - 3)}{10} \) by dividing both the numerator and denominator by 2, resulting in \( \frac{4n - 3}{5} \).

Key Concepts

Simplifying Fractions

When you're working with fractions, it's often useful to simplify them. Simplifying fractions means reducing them to their smallest possible form. This is done by finding a number that both the numerator and denominator can be divided by. In our case, \[ \frac{8n - 6}{10} \] both 8 and 6 in the numerator, as well as the 10 in the denominator, are divisible by 2. By factoring out 2 from the numerator, we'll have:

• Numerator: \( 2(4n - 3) \)
• Denominator: 10

Next, divide both the numerator and the denominator by 2, leading to:\[ \frac{4n - 3}{5} \]It's important always to check if a fraction can be simplified to make it easier to work with in future calculations.

Common Denominator

When adding or subtracting fractions, having a common denominator is essential. The denominator is the bottom number of a fraction. For operations like addition and subtraction of fractions, a common denominator allows you to directly add or subtract the numerators. In our exercise, both fractions \( \frac{2n-5}{10} \) and \( \frac{6n-1}{10} \) already share a common denominator of 10.

• This means we can immediately combine the fractions.
• No need to adjust either fraction, making the process quicker.

After ensuring the denominators are the same, you can focus on handling the numerators confidently.

Numerator and Denominator

Understanding numerators and denominators is key to mastering fractions. The numerator is the top part of a fraction that indicates how many parts of the whole are being considered. The denominator, on the bottom, shows the total number of equal parts the whole is divided into. To add fractions like \( \frac{2n-5}{10} \) and \( \frac{6n-1}{10} \), we:

• Add the numerators: \((2n - 5) + (6n - 1)\)
• Keep the denominator the same: 10

This results in the new fraction:\[ \frac{8n - 6}{10} \]Understanding these roles helps in performing arithmetic operations on fractions effortlessly.

Algebraic Fractions

Algebraic fractions involve variables in the numerator, denominator, or both. They function like regular fractions but may require factoring or expanding the terms involved. In our example, \( \frac{2n-5}{10} \) and \( \frac{6n-1}{10} \), the variable \(n\) appears in the numerator, making these algebraic fractions. Here are steps to handle them effectively:

• Treat them as regular fractions for addition or subtraction.
• Simplify by combining like terms—in this case, \((2n + 6n)\) becomes \(8n\).
• Simplify further by reducing the entire fraction, as seen in our final form, \(\frac{4n - 3}{5}\).

Handling algebraic fractions becomes much easier with practice as you become more familiar with factoring and identifying common terms.