Problem 90 - Elementary Algebra

Question

Simplify each algebraic expression by combinang similar terms. $$2 n-\frac{6}{7} n+\frac{5}{14} n$$

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{3}{2}n\).

Step 1: Identify Similar Terms

The expression is composed of terms with the variable \(n\). Therefore, we need to combine: \(2n\), \(-\frac{6}{7}n\), and \(\frac{5}{14}n\) as they are all like terms with the same base and exponent.

Step 2: Find a Common Denominator

The coefficients are \(2\), \(-\frac{6}{7}\), and \(\frac{5}{14}\). To combine them, let's first find a common denominator for the fractional terms \(-\frac{6}{7}\) and \(\frac{5}{14}\). The common denominator between 7 and 14 is 14.

Step 3: Convert to Common Denominator

Convert each fraction to have a denominator of 14: \(-\frac{6}{7} = -\frac{12}{14}\) because \(-6\times2 = -12\) and \(7\times2 = 14\). Now rewrite each term as \(2 = \frac{28}{14}\) (since \(2\times14 = 28\)). The terms are now \(\frac{28}{14}n\), \(-\frac{12}{14}n\), and \(\frac{5}{14}n\).

Step 4: Combine the Terms

Add the coefficients together now that they have the same denominator: \(\frac{28}{14} - \frac{12}{14} + \frac{5}{14} = \frac{21}{14}\).

Step 5: Simplify the Fraction

Simplify \(\frac{21}{14}\) by dividing the numerator and the denominator by their greatest common divisor, which is 7: \(\frac{21}{14} = \frac{21\div7}{14\div7} = \frac{3}{2}\).

Step 6: Write the Final Expression

The simplified algebraic expression is \(\frac{3}{2}n\) after combining similar terms and simplifying.

Key Concepts

Combining Like Terms

Algebraic expressions often contain terms that are similar. These similar terms are known as "like terms." They have the same variables raised to the same power. In the expression \(2n - \frac{6}{7}n + \frac{5}{14}n\), all terms are like terms because they have the variable \(n\) and the power for \(n\) in each term is 1.
To simplify an expression, you combine like terms. Combining means adding or subtracting their coefficients. For instance, if you have \(3n + 4n\), you can combine them to \(7n\) because both terms have the same variable and exponent. This process helps in simplifying expressions and makes solving algebraic equations much easier.
When you see terms with the same variable and exponent, remember they can be combined by simply adding or subtracting the numbers in front of the variables, as these numbers, known as coefficients, guide how much of the variable is in each term.

Common Denominator

When dealing with fractions, finding a common denominator is crucial to perform addition or subtraction. A denominator is the bottom part of a fraction, and a common denominator is a shared multiple of the denominators of several fractions. For the expression \(2n - \frac{6}{7}n + \frac{5}{14}n\), you deal with fractional coefficients, specifically \(-\frac{6}{7}\) and \(\frac{5}{14}\).
Before you can combine these fractions, it's necessary to convert them so that they share the same denominator. Identify the least common multiple (LCM) of the denominators 7 and 14, which in this case is 14. You will rewrite each fraction with this common denominator to ensure they can be easily added or subtracted.
To adjust each fraction to have the common denominator of 14, consider:

• For \(-\frac{6}{7}\), multiply both the numerator and denominator by 2 to get \(-\frac{12}{14}\).
• For \(\frac{5}{14}\), it already has 14 as its denominator, so no change is needed.

Finding a common denominator allows smooth operations between fractions, simplifying complex expressions.

Simplifying Fractions

Simplifying fractions is a pivotal step in reducing algebraic expressions to their simplest form. It involves making the fraction as simple as possible by ensuring the numerator and the denominator have no common factors besides 1. This process is evident in the amalgamation of the terms in the given exercise.
After combining the like terms \(\frac{28}{14}n - \frac{12}{14}n + \frac{5}{14}n\), you obtain the result \(\frac{21}{14}n\). To simplify, find the greatest common divisor (GCD) of 21 and 14, which is 7. You then divide both the numerator and the denominator by this GCD:
\[\frac{21}{14} = \frac{21 \div 7}{14 \div 7} = \frac{3}{2}\]Thus, the original expression \(2n - \frac{6}{7}n + \frac{5}{14}n\) simplifies to \(\frac{3}{2}n\).
Breaking down complex fractions through simplification results in more manageable and interpretable expressions, facilitating clearer and quicker understanding of algebraic problems.