Problem 5 - Intermediate Algebra

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 variable term

Start by isolating the term containing the variable on one side of the equation. For the given equation \( \frac{n}{2} - \frac{2}{3} = \frac{5}{6} \), add \( \frac{2}{3} \) to both sides to isolate \( \frac{n}{2} \) on the left side:\[\frac{n}{2} - \frac{2}{3} + \frac{2}{3} = \frac{5}{6} + \frac{2}{3}\]This simplifies to: \[\frac{n}{2} = \frac{5}{6} + \frac{2}{3}\]

Step 2: Find a common denominator

To add \( \frac{5}{6} \) and \( \frac{2}{3} \), find a common denominator. The least common denominator for 6 and 3 is 6. Convert \( \frac{2}{3} \) to have 6 as the denominator:\[\frac{2}{3} = \frac{2\times2}{3\times2} = \frac{4}{6}\]

Step 3: Add the fractions

Add the fractions on the right side of the equation:\[\frac{5}{6} + \frac{4}{6} = \frac{9}{6}\]Thus, the simplified equation becomes:\[\frac{n}{2} = \frac{9}{6}\]

Step 4: Simplify the fraction

Simplify \( \frac{9}{6} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 3:\[\frac{9}{6} = \frac{9\div3}{6\div3} = \frac{3}{2}\]So, the equation now reads:\[\frac{n}{2} = \frac{3}{2}\]

Step 5: Solve for the variable

To find \( n \), multiply both sides of the equation by 2:\[2 \times \frac{n}{2} = 2 \times \frac{3}{2}\]This simplifies to:\[n = 3\]

Key Concepts

Solving Equations

Solving equations is like solving a puzzle. You need to find the value of the variable that makes the equation true. In this example, the variable is \(n\). The equation given is \(\frac{n}{2} - \frac{2}{3} = \frac{5}{6}\). The first step is to isolate the variable term. This means getting the part of the equation containing \(n\) on one side. We achieve this by performing the same operation on both sides of the equation.
For the given equation, we add \(\frac{2}{3}\) to both sides.

• This results in \(\frac{n}{2} = \frac{5}{6} + \frac{2}{3}\).

By isolating the variable term and maintaining balance in the equation, one can simplify the process of solving for the unknown.

Fraction Operations

Working with fractions is an important part of mathematics. To add or subtract fractions, they must have the same denominator. In this case, we need to add \(\frac{5}{6}\) and \(\frac{2}{3}\). First, identify the least common denominator (LCD), which is the smallest number that both denominators can divide into without a remainder.
For 6 and 3, the LCD is 6. This means we need to convert \(\frac{2}{3}\) to have 6 as a denominator:

• Multiply both the numerator and denominator by 2 to get \(\frac{4}{6}\).
• Now we can easily add the fractions: \(\frac{5}{6} + \frac{4}{6} = \frac{9}{6}\).

Simplifying fractions helps in making calculations easier and clearer, which leads to solving equations more efficiently.

Algebraic Manipulation

Algebraic manipulation involves transforming an equation to make it easier to solve. Once we have \(\frac{n}{2} = \frac{9}{6}\) from the previous operations, the next step is simplification and solving for \(n\). Start by simplifying the fraction \(\frac{9}{6}\):

• The greatest common divisor of 9 and 6 is 3.
• Divide both numerator and denominator by 3 to simplify it to \(\frac{3}{2}\).

Now, we have \(\frac{n}{2} = \frac{3}{2}\). To solve for \(n\), multiply both sides by 2.

• This will result in \(n = 3\), which is the solution.

Using algebraic manipulation allows us to handle complex expressions and solve equations efficiently, ultimately finding the solution.