Problem 22 - Elementary Algebra

Question

Add or subtract as indicated. Be sure to express your answers in simplest forn. (Objective 1) $$\frac{2 n-6}{5}-\frac{7 n-1}{5}$$

Step-by-Step Solution

Answer

The simplified result is \(-n - 1\).

Step 1: Identify Common Denominator

Notice that both fractions \(\frac{2n-6}{5}\) and \(\frac{7n-1}{5}\) have the same denominator, which is 5. This means that we can directly subtract the numerators while keeping the same denominator.

Step 2: Subtract the Numerators

Subtract the numerators of the two fractions. The numerator of the first fraction is \(2n - 6\) and the numerator of the second fraction is \(7n - 1\). So, the subtraction is:\[(2n - 6) - (7n - 1)\]

Step 3: Simplify the Subtraction of Numerators

Distribute the negative sign and simplify:\[2n - 6 - 7n + 1 = (2n - 7n) + (-6 + 1)\]This results in:\[-5n - 5\]

Step 4: Form the Resulting Fraction

Combine the simplified numerator with the common denominator to complete the fraction:\[\frac{-5n - 5}{5}\]

Step 5: Simplify the Fraction

Factor out a common factor from the numerator, which is -5, to simplify:\[\frac{-5(n + 1)}{5}\]Cancel out the common factor of 5 in the numerator and denominator to get:\[-(n + 1) = -n - 1\]

Key Concepts

Fraction Operations

Working with fractions can be simple once you understand the basic operations involved. When you add or subtract fractions, ensure the denominators (the number at the bottom) are the same. If they’re not the same, you’ll need to find the least common multiple to make them match. In our example, \(\frac{2n-6}{5}\) and \(\frac{7n-1}{5}\) already share the same denominator, which is 5. This means we can move on directly to subtracting the numerators.

Here are some steps to keep in mind when handling fraction operations:

• Check if denominators are the same.
• If they are, subtract or add the numerators as needed.
• Keep the common denominator.
• Once combined, simplify the fraction if possible.

Always ensure your final answer is in its simplest form, which involves checking if the numerator and denominator share any common factors.

Simplifying Expressions

Simplifying expressions is crucial to making mathematics easier to work with and understand. Begin by resolving parentheses and exponents, then move on to any addition, subtraction, multiplication, and division. You should simplify expressions by grouping like terms and reducing constants.

In the example provided, after subtracting the numerators, you were left with:\[2n - 6 - 7n + 1\] which simplifies to: \[-5n - 5\].
To simplify further, factor out the common factor in the numerator. Notice that both terms contain a factor of -5:\[-5(n + 1)\].

The objective is to reach the simplest form possible by eliminating any factors common to the numerator and denominator.

Common Denominators

Understanding common denominators is a fundamental part of manipulating fractions in algebra. A common denominator allows you to add or subtract properly by aligning the expressions.
Denominators are the numbers beneath the fraction line. For the two fractions \(\frac{2n-6}{5}\) and \(\frac{7n-1}{5}\),
the denominator 5 is already common. This allows us to subtract the numerators without further modifications to the denominators.

Consider these steps if you encounter fractions with different denominators:

• Identify the least common multiple (LCM) of the denominators.
• Adjust each fraction by multiplying both its numerator and denominator to obtain the LCM as the new denominator.
• Once the fractions share a common denominator, perform the addition or subtraction.

Maintaining a common denominator simplifies the process and ensures accuracy when combining fractions.