Problem 21 - Elementary Algebra

Question

Add or subtract as indicated. Be sure to express your answers in simplest forn. (Objective 1) $$\frac{3 n-7}{6}-\frac{9 n-1}{6}$$

Step-by-Step Solution

Answer

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

Step 1: Understand the Common Denominator

Notice that both fractions have the same denominator. This means that we can combine the numerators directly, as the denominator does not change.

Step 2: Subtract the Numerators

Since the operation is a subtraction, subtract the second numerator from the first:\[(3n - 7) - (9n - 1)\]Simplify this expression by distributing the subtraction across the terms inside the parentheses:\[3n - 7 - 9n + 1\]

Step 3: Simplify the Expression

Combine like terms in the expression:\[(3n - 9n) + (-7 + 1)\]This simplifies to:\[-6n - 6\]

Step 4: Combine with the Common Denominator

Now put the simplified numerator back over the common denominator:\[\frac{-6n - 6}{6}\]

Step 5: Simplify the Fraction

Factor out the greatest common factor in the numerator, which is 6:\[\frac{6(-n - 1)}{6}\]Now cancel the common factor of 6 in the numerator and denominator:\[-n - 1\]

Step 6: Express the Final Simplified Result

The result of the operation in its simplest form is:\[-n - 1\]

Key Concepts

Common Denominator

A common denominator is essential when adding or subtracting fractions. It allows us to combine the fractions without changing their values. In this exercise, the fractions have a common denominator of 6, which simplifies the process.

When fractions share the same denominator, you can work directly with the numerators. This is because the baseline for comparison, the denominator, is already uniform.

Here's a quick tip to identify common denominators:

• If the denominators are the same, you have a common denominator.
• If they are different, find the least common multiple of the denominators.

Starting with the same denominator, as in our problem, simplifies the procedure significantly. It allows for immediate attention to the numerators without further alteration.

Simplifying Expressions

Once you've aligned the fractions over a common denominator, the next step is simplifying the expression. Simplifying involves breaking down an expression into its most basic form. This often includes removing parentheses and combining like terms.

In the exercise, you subtract the numerators after applying the subtraction to each term: subtract the whole second expression from the first. "(3n - 7) - (9n - 1)" becomes "3n - 7 - 9n + 1". Here, we've distributed the negative sign across the terms in the parentheses.

Simplification helps in making complex expressions more manageable and clearer. As you become proficient, you'll recognize patterns, making this step swift and intuitive.

Combining Like Terms

Combining like terms is about simplification. It involves grouping terms in an expression that have the same variable and power. This process makes an expression easier to understand and use.

In our example, the terms "3n" and "-9n" are like terms because they both involve the variable "n". Similarly, "-7" and "+1" are constants.

• Combine the coefficients of like terms: (3n - 9n) results in -6n.
• And for constants: (-7 + 1) results in -6.

This results in the simplified expression: "-6n - 6". A clean, simplified expression is crucial for clarity and accurate solving in algebra. It also assists in quickly identifying further simplifications or the final solution.