Problem 6 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The solution to the equation is \(n = 5\).

Step 1: Eliminate Fractions

To make the solving process simpler, eliminate the fractions by finding the least common denominator (LCD) of all fractions involved. The fractions are \(\frac{n}{4}, \frac{5}{6},\) and \(\frac{5}{12}\). The LCD of 4, 6, and 12 is 12. Multiply every term of the equation by 12: \[12 \left( \frac{n}{4} \right) - 12 \left( \frac{5}{6} \right) = 12 \left( \frac{5}{12} \right) \] Which simplifies to:\[3n - 10 = 5\]

Step 2: Isolate the Variable Term

Next, move the constant term \(-10\) on the left side to the right side by adding 10 to both sides of the equation:\[3n - 10 + 10 = 5 + 10\]\[3n = 15\]

Step 3: Solve for Variable

To solve for \(n\), divide each side of the equation \(3n = 15\) by 3:\[\frac{3n}{3} = \frac{15}{3}\]\[n = 5\]

Key Concepts

Least Common Denominator

Understanding the least common denominator (LCD) is crucial when dealing with fractions in algebra. The LCD is the smallest shared multiple of the denominators in the fractions you are working with. It allows you to eliminate the fractions and thus simplify the equation-solving process.

To find the LCD, list the multiples of each denominator until you find the smallest common one.

• For 4: 4, 8, 12, 16...
• For 6: 6, 12, 18, 24...
• For 12: 12, 24, 36...

In the equation \(\frac{n}{4} - \frac{5}{6} = \frac{5}{12}\), the LCD of 4, 6, and 12 is 12. This process ensures that the fractions have a common denominator, which helps us eliminate them in the next steps.

Fractions in Algebra

Fractions can seem tricky, especially in algebraic equations, but with a clear approach, they become much easier to manage. When an equation involves fractions, handling them correctly is essential for finding the solution. Here are some important points to consider:

• Identify all fractions involved in the equation.
• Find a common denominator to standardize the fractions, as this simplifies the addition, subtraction, or comparison.
• Multiply each fraction by this number to eliminate the denominators from the equation entirely.

In our example, after identifying the common denominator, each term was multiplied by 12. This step removes the fractions and turns the equation into one with whole numbers, allowing for straightforward algebraic manipulation.

Step-by-Step Equation Solving

Solving equations step-by-step helps in understanding the logic and method of isolating the variable to find its value. This methodical approach is beneficial, especially when variables and constants are mixed in the equation. Let's see how it's done:

1. **Eliminate Fractions**: - Multiply every term by the LCD to get rid of the denominators. - In our equation, multiplying by 12 simplifies it to \(3n - 10 = 5\).
2. **Isolate the Variable**: - Shift terms to ensure the variable is on one side. Performing operations like addition or subtraction on both sides ensures the balance. - Here, adding 10 to both sides isolates the variable term as \(3n = 15\).
3. **Solve for the Variable**: - Divide to find the variable's value, ensuring that you perform the same operation across the equation. - Divide both sides by 3 to get \(n = 5\).This logical sequence of steps guarantees that you systematically solve for the unknown without errors.