Problem 6 - Algebra for College Students

Question

Solve each equation. \(\frac{n}{4}-\frac{5}{6}=\frac{5}{12}\)

Step-by-Step Solution

Answer

The solution is \( n = 5 \).

Step 1: Eliminate Fractions by Finding the Least Common Denominator

The given equation is \( \frac{n}{4} - \frac{5}{6} = \frac{5}{12} \). Identify the least common denominator (LCD) of the fractions involved, which are 4, 6, and 12. The LCD is 12. Multiply each term in the equation by 12 to eliminate the fractions:\[ 12 \left( \frac{n}{4} \right) - 12 \left( \frac{5}{6} \right) = 12 \left( \frac{5}{12} \right) \]

Step 2: Simplify the Equation

Apply the multiplication to each term:- \( 12 \times \frac{n}{4} = 3n \)- \( 12 \times \frac{5}{6} = 10 \)- \( 12 \times \frac{5}{12} = 5 \)This simplifies the equation to:\[ 3n - 10 = 5 \]

Step 3: Isolate the Variable Term

To isolate \( 3n \), add 10 to both sides of the equation:\[ 3n - 10 + 10 = 5 + 10 \]Simplifying gives:\[ 3n = 15 \]

Step 4: Solve for the Variable

Solve for \( n \) by dividing both sides of the equation by 3:\[ \frac{3n}{3} = \frac{15}{3} \]This results in:\[ n = 5 \]

Step 5: Check the Solution

Substitute \( n = 5 \) back into the original equation to verify:\[ \frac{5}{4} - \frac{5}{6} = \frac{5}{12} \]Find a common denominator for \( \frac{5}{4} \) and \( \frac{5}{6} \), which is 12:- Convert \( \frac{5}{4} \) to \( \frac{15}{12} \)- Convert \( \frac{5}{6} \) to \( \frac{10}{12} \)Calculate:\[ \frac{15}{12} - \frac{10}{12} = \frac{5}{12} \]The left side equals the right side, confirming that \( n = 5 \) is correct.

Key Concepts

Least Common Denominator

When solving equations with fractions, one of the first steps is to eliminate the fractions to make the equation easier to handle. The goal is to find the Least Common Denominator (LCD), which is the smallest number that all the denominators divide into evenly. In our given equation, the fractions have denominators 4, 6, and 12. To find the LCD of these numbers:

• List the multiples of each denominator until you find the smallest common multiple
• For 4: 4, 8, 12, 16...
• For 6: 6, 12, 18...
• For 12: 12, 24, 36...

The smallest number that appears in all lists is 12, making it the LCD. Multiplying each term by this LCD gets rid of the fractions and simplifies the equation. For instance, multiplying each fraction in the equation by 12 means effectively multiplying both sides of the equation by the same number, ensuring the equation remains balanced.

Simplifying Equations

Once the Least Common Denominator has removed fractions from the equation, the next goal is to simplify. Simplifying involves straightforward arithmetic calculations that reduce the equation to its simplest form. After multiplying each term of the equation by 12, as in our example, you need to compute each term individually:

• The term \( \frac{n}{4} \times 12 \) simplifies to \( 3n \) because \( 12 \div 4 = 3 \)
• For \( - \frac{5}{6} \times 12 \), it becomes \( -10 \) since \( 12 \div 6 = 2 \) and \( 5 \times 2 = 10 \)
• The fraction \( \frac{5}{12} \times 12 \) simplifies to \( 5 \) as \( 12 \div 12 = 1 \)

By simplifying each term, you transform the complex equation into a simpler one: \( 3n - 10 = 5 \). This makes it much more manageable to solve.

Isolating the Variable

The purpose of isolating the variable is to find its value, which is the ultimate goal of solving an equation. Once you have a simplified linear equation, like \( 3n - 10 = 5 \), start by getting the variable term by itself on one side of the equation. This involves basic algebraic techniques:

• First, add or subtract numbers from both sides to remove any constants next to the variable. In this example, add 10 to both sides to get \( 3n = 15 \).

The next step is to "undo" the multiplication that is presently keeping the variable term attached with other numbers. Here, divide both sides by 3, resulting in \( n = 5 \). Isolating the variable effectively gives you its value and allows you to check your work.

Checking Solutions in Algebra

After finding a solution, it's crucial to verify its accuracy. This involves substituting the found value back into the original equation to ensure both sides are equal. For our solved equation, we substitute \( n = 5 \) back into the original fractions:

• The term becomes \( \frac{5}{4} \)
• Subtract \( \frac{5}{6} \) from \( \frac{5}{4} \)

Since these are fractions, find a common denominator, in this case, 12:

• Convert \( \frac{5}{4} \) to \( \frac{15}{12} \)
• Convert \( \frac{5}{6} \) to \( \frac{10}{12} \)
• Then calculate \( \frac{15}{12} - \frac{10}{12} = \frac{5}{12} \)

Both sides match, confirming \( n = 5 \) as correct. This final check ensures your solution is truly valid and strengthens your understanding of solving linear equations with fractions.