Problem 89 - Elementary Algebra

Question

Simplify each algebraic expression by combinang similar terms. $$n+\frac{4}{3} n-\frac{1}{9} n$$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{20}{9}n \).

Step 1: Identify Like Terms

Start by identifying all the like terms in the expression. The expression given is \( n + \frac{4}{3}n - \frac{1}{9}n \). All these terms have a common variable part \( n \). Therefore, they are like terms and can be combined.

Step 2: Make a Common Denominator

Before combining the coefficients, ensure that all fractions have a common denominator. In this expression, 1 is also a fraction (\(1 = \frac{3}{3}\)). So, the fractions become \( \frac{3}{3}n, \frac{4}{3}n, \text{ and } -\frac{1}{9}n \). The least common multiple of 3 and 9 is 9. Convert all the terms to have a denominator of 9.

Step 3: Convert to Common Denominator

Convert all coefficients to have a denominator of 9:- \(\frac{3}{3}n\) becomes \(\frac{9}{9}n\)- \(\frac{4}{3}n\) becomes \(\frac{12}{9}n\)- \(-\frac{1}{9}n\) stays as \(-\frac{1}{9}n\)

Step 4: Combine the Fractions

Add the fractions: \[ \frac{9}{9}n + \frac{12}{9}n - \frac{1}{9}n \]Combine the numerators over the common denominator:\[ \left(9 + 12 - 1\right)\frac{n}{9} = \frac{20}{9}n \]

Step 5: Simplify the Expression

The final simplified expression is \( \frac{20}{9}n \). Since 20 and 9 have no common factors, this fraction is already in its simplest form.

Key Concepts

Combining Like Terms

When simplifying algebraic expressions, the first thing to do is identify and combine like terms. Like terms are terms in an expression that have the same variable raised to the same power. In the expression given, \( n + \frac{4}{3}n - \frac{1}{9}n \), all the terms have the variable \( n \). This means they are like terms.
Combining like terms is a crucial step as it simplifies the expression, making it easier to work with. To do so, we need to add or subtract the coefficients of these like terms. Coefficients are the numerical parts in front of the variables, like 1, \( \frac{4}{3} \), and \( -\frac{1}{9} \) in our problem. Simplifying involves adding these coefficients together while keeping the common variable part.

Common Denominators

When combining terms that include fractions, we need to work with common denominators. A common denominator is a shared multiple of the denominators involved. In our expression, the fractions are \( \frac{4}{3}n \) and \( -\frac{1}{9}n \).

• The denominators here are 3 and 9.
• The least common multiple (LCM) of these numbers is 9.

This means we need to convert fractions to equivalent fractions with a denominator of 9. For \( \frac{4}{3}n \), we multiply both the numerator and denominator by 3 to get \( \frac{12}{9}n \). For any single term with \( n \) or equivalent integer, we express it as a fraction with the relevant common denominator, like \( \frac{9}{9}n \).

Fraction Simplification

Once all terms have the same denominator, the task shifts to combining the numerators. This is the exciting part of fraction simplification, where the process becomes straightforward. With the terms \( \frac{9}{9}n \), \( \frac{12}{9}n \), and \( -\frac{1}{9}n \), the numerators (9, 12, and -1) are combined:
The operation simple becomes:

• Add 9 and 12, which equal 21.
• Subtract 1 from 21 to get 20.

This results in the fraction \( \frac{20}{9}n \). Fraction simplification emphasizes keeping the denominator the same while only combining the numerators. This systematic approach ensures clarity and accuracy in simplifying.

Algebraic Fractions

Algebraic fractions feature variables in the numerator, the denominator, or both. Our job is to simplify them to aid in calculations and further algebraic analysis. They follow the same rules as numeric fractions, requiring:

• Common denominators for addition or subtraction.
• Keeping expressions in equivalent forms.

In the exercise discussed, each fraction carried the same variable part \( n \) which assisted in simplifying via algebraic principles. While the math involved seems simple, the focus is maintained on accuracy, as small mistakes in handling algebraic fractions can lead to erroneous conclusions. With \( \frac{20}{9}n \), we've reached the simplest form for this expression, ensuring it's ready for any further algebraic manipulations or computations.