Problem 5 - Elementary Algebra

Question

\(\frac{n}{2}+\frac{n-1}{6}=\frac{5}{2}\)

Step-by-Step Solution

Answer

The solution for \(n\) is 4.

Step 1: Find a common denominator

The denominators in the equation are 2, 6, and 2. The least common multiple of these numbers is 6. Therefore, we need to express all fractions with a denominator of 6.

Step 2: Rewrite each term with the common denominator

Convert each fraction to have a denominator of 6: - \(\frac{n}{2}\) becomes \(\frac{3n}{6}\), because \(n/2 = 3n/6\).- \(\frac{n-1}{6}\) is already with a denominator 6, so we rewrite it as is.- \(\frac{5}{2}\) becomes \(\frac{15}{6}\), because \(5/2 = 15/6\).

Step 3: Combine the fractions on the left-hand side

Now, add the fractions \(\frac{3n}{6} + \frac{n-1}{6}\) on the left-hand side:\[\frac{3n + n - 1}{6} = \frac{4n - 1}{6}\]

Step 4: Set the equation with common denominators

Now the equation is: \[\frac{4n - 1}{6} = \frac{15}{6}\] Since the denominators are the same, we focus on the numerators.

Step 5: Solve for the numerator

Since the denominators are equal, set the numerators equal to each other:\[4n - 1 = 15\].Add 1 to both sides to isolate terms with \(n\):\[4n = 16\]

Step 6: Solve for \(n\)

Divide both sides by 4 to solve for \(n\):\[n = \frac{16}{4} = 4\]

Key Concepts

Understanding the Common Denominator

A common denominator is essential when adding or comparing fractions, especially in algebra equations. Essentially, it is a shared multiple of the denominators you have in your equation. This makes it easier to combine and compare different fractions since they will all share the same base.
Understanding why we need a common denominator:

• When fractions have different denominators, they represent parts of different wholes, making it impossible to straightforwardly add or subtract them.
• A common denominator standardizes these fractions, allowing us to process and manipulate them mathematically.

To find a common denominator, you usually determine the least common multiple (LCM) of all denominators involved. In our exercise, the denominators are 2, 6, and 2. The LCM is 6, which becomes our common denominator. Rewriting each fraction with this common base ensures accurate and versatile calculations.

Simplifying Fractions

Simplifying fractions is a process of reducing a fraction to its simplest form or leveraging a common denominator to combine fractions. In this exercise, the fractions were rewritten to have the same denominator, enabling us to add or compare them directly:
Steps to simplify fractions:

• Convert each fraction to share a common denominator. For example, in this equation, the fraction \(\frac{n}{2}\) becomes \(\frac{3n}{6}\).
• Add or subtract the numerators while maintaining the common denominator. This involves operating on the numerator parts of the equation, like \(\frac{3n + n - 1}{6}\) simplifies to \(\frac{4n - 1}{6}\).

By simplifying fractions, particularly with a common denominator, equations become easier to solve or rearrange. The approach cuts down complex operations into manageable, step-by-step calculations.

Solving Linear Equations

Linear equations are equations where the highest power of the variable is one. They are straightforward to solve once you establish a consistent method. After establishing a common denominator and simplifying the fractions, solving the equation boils down to balancing the equation by manipulating these fractions or sums to isolate the variable.
Steps to solve the provided linear equation:

• First, rewrite the equation with common denominators: \(\frac{4n - 1}{6} = \frac{15}{6}\).
• Equate the numerators, because the denominators are already equal: \(4n - 1 = 15\).
• Complete the operations by isolating the variable, \(n\): add or subtract constants first (add 1 to both sides, so \(4n = 16\)).
• Finally, solve for \(n\) by dividing both sides by 4, resulting in \(n = 4\).

Linear equations, especially with variables in fractions, require clear and orderly steps. Each step effectively simplifies the equation, revealing the value of the unknown variable. This approach makes these problems accessible and logic-driven.