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Magazine Chapter 2: Problem 5
Solve each equation. \(\frac{n}{2}-\frac{2}{3}=\frac{5}{6}\)
Short Answer
Expert verified
The solution is \( n = 3 \).
Step by step solution
01
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: \( \frac{n}{2} = \frac{5}{6} + \frac{2}{3} \).
02
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: \( \frac{5}{6} + \frac{4}{6} = \frac{9}{6} \).
03
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 \( \frac{n}{2} = \frac{3}{2} \).
04
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: \( n = 3 \).
05
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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: \( \frac{9}{6} \).
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 \( -\frac{2}{3} \).
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: \( \frac{n}{2} = \frac{5}{6} + \frac{2}{3} \).
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.
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: \( \frac{3}{2} \).
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:
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: \( \frac{9}{6} - \frac{4}{6} = \frac{5}{6} \)
