Problem 5 - Algebra for College Students

Question

Solve each equation. \(\frac{n}{2}-\frac{2}{3}=\frac{5}{6}\)

Step-by-Step Solution

Answer

The solution is \( n = 3 \).

Step 1: Isolate the term with the variable

We start with the equation \( \frac{n}{2} - \frac{2}{3} = \frac{5}{6} \). To isolate the term with the variable \( \frac{n}{2} \), we add \( \frac{2}{3} \) to both sides of the equation: \( \frac{n}{2} = \frac{5}{6} + \frac{2}{3} \).

Step 2: Add fractions on the right side

Next, we need to add \( \frac{5}{6} \) and \( \frac{2}{3} \). To add these fractions, find a common denominator. The least common denominator for 6 and 3 is 6. Rewrite \( \frac{2}{3} \) as \( \frac{4}{6} \). Now add the fractions: \( \frac{5}{6} + \frac{4}{6} = \frac{9}{6} \).

Step 3: Simplify the fraction

The fraction \( \frac{9}{6} \) can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3. So, \( \frac{9}{6} = \frac{3}{2} \). Thus, the equation becomes \( \frac{n}{2} = \frac{3}{2} \).

Step 4: Solve for the variable

To solve for \( n \), multiply both sides of the equation \( \frac{n}{2} = \frac{3}{2} \) by 2 to get rid of the fraction: \( n = 3 \).

Step 5: Verify the solution

Substitute \( n = 3 \) back into the original equation to verify the solution: \( \frac{3}{2} - \frac{2}{3} \). Convert both fractions to have a common denominator, \( \frac{3}{2} = \frac{9}{6} \) and \( \frac{2}{3} = \frac{4}{6} \). Then, \( \frac{9}{6} - \frac{4}{6} = \frac{5}{6} \), which is the right side of the original equation, thus confirming our solution.

Key Concepts

Adding Fractions

When adding fractions, it's crucial that they have a common denominator. This allows the fractions to be combined directly since the denominators represent the same value.
To find a common denominator, consider the smallest number that both denominators can divide evenly into, known as the least common denominator (LCD).
For instance, to add \( \frac{5}{6} \) and \( \frac{2}{3} \), the denominators 6 and 3 both share a least common multiple of 6.

• Rewrite \( \frac{2}{3} \) as \( \frac{4}{6} \) so both fractions share the denominator 6.
• Add the numerators: \( 5 + 4 = 9 \), placing the sum over the common denominator: \( \frac{9}{6} \).

Once combined, you may need to simplify the fraction.

Isolating Variables

Isolating the variable means getting the variable on one side of the equation by itself. Let's look at the equation \( \frac{n}{2} - \frac{2}{3} = \frac{5}{6} \).
The goal is to isolate \( \frac{n}{2} \) by eliminating the constant term \( -\frac{2}{3} \).

• Add \( \frac{2}{3} \) to both sides of the equation.
• This step yields: \( \frac{n}{2} = \frac{5}{6} + \frac{2}{3} \).

By isolating the variable step by step, you simplify the equation for easier manipulation, ultimately solving for the variable's value.

Simplifying Fractions

Simplifying fractions means reducing them to their lowest terms. This is done by dividing the numerator and the denominator by their greatest common divisor. Take \( \frac{9}{6} \) as an example.
Here, both 9 and 6 can be divided by 3, the greatest common divisor.

• Divide the numerator: \( 9 \div 3 = 3 \).
• Divide the denominator: \( 6 \div 3 = 2 \).
• Resulting in the simplified fraction: \( \frac{3}{2} \).

Simplifying ensures fractions are in their most compact form, often making equations easier to solve.

Verifying Solutions

Verifying solutions involves plugging the found variable back into the original equation to ensure correctness. When \( n = 3 \) is inserted back into \( \frac{n}{2} - \frac{2}{3} = \frac{5}{6} \), you re-calculate to see if both sides match.
Replace \( n \) with 3: \( \frac{3}{2} - \frac{2}{3} \). Convert them to a common denominator:

• \( \frac{3}{2} = \frac{9}{6} \)
• \( \frac{2}{3} = \frac{4}{6} \)
• Perform the subtraction: \( \frac{9}{6} - \frac{4}{6} = \frac{5}{6} \)

Since they match the right side of the original equation, the solution is verified and correct.