Problem 27 - Algebra for College Students

Question

Solve each equation. \(\frac{2 a-3}{6}+\frac{3 a-2}{4}+\frac{5 a+6}{12}=4\)

Step-by-Step Solution

Answer

The solution to the equation is \( a = 3 \).

Step 1: Identify the Least Common Denominator (LCD)

The fractions in the equation have denominators 6, 4, and 12. To clear the fractions, we need the least common denominator. The least common multiple of 6, 4, and 12 is 12.

Step 2: Eliminate Fractions

Multiply every term in the equation by 12, the LCD, to eliminate fractions: \[12 \left( \frac{2a-3}{6} \right) + 12 \left( \frac{3a-2}{4} \right) + 12 \left( \frac{5a+6}{12} \right) = 12 \times 4 \] Simplifying each term: \[ 2 \times (2a-3) + 3 \times (3a-2) + 1 \times (5a+6) = 48 \] This leads to: \[ 4a - 6 + 9a - 6 + 5a + 6 = 48 \]

Step 3: Combine Like Terms

Combine the like terms in the equation: \[ (4a + 9a + 5a) + (-6 - 6 + 6) = 48 \] Which simplifies to: \[ 18a - 6 = 48 \]

Step 4: Isolate the Variable

First, add 6 to both sides of the equation to remove the constant on the left side: \[ 18a - 6 + 6 = 48 + 6 \] Simplifying gives: \[ 18a = 54 \]

Step 5: Solve for 'a'

Divide both sides by 18 to solve for 'a': \[ a = \frac{54}{18} \] Simplifying the fraction gives: \[ a = 3 \]

Key Concepts

Least Common Denominator

The concept of the Least Common Denominator (LCD) is crucial when dealing with equations containing fractions. The LCD is the smallest number that each of the denominators in your equation can divide into equally. By determining the LCD, you can simplify and solve fractional equations more effectively.
For example, consider the denominators in the equation: 6, 4, and 12. To find the LCD, identify the least common multiple (LCM) of these numbers.

• List the multiples of each number:
  • Multiples of 6: 6, 12, 18, 24,...
  • Multiples of 4: 4, 8, 12, 16,...
  • Multiples of 12: 12, 24, 36,...
• The lowest common multiple shared by these numbers is 12.

Utilizing the LCD allows you to convert all fractions to have a common denominator, making the subsequent steps, such as elimination, much easier.

Fraction Elimination

Once the Least Common Denominator is determined, you can proceed to the Fraction Elimination stage. This step is all about clearing the fractions from your equation to simplify the process of solving it.
To eliminate fractions, multiply every term in your equation by the LCD (in this case, 12). This transforms each fractional term into a whole number expression:

• \[12 \times \left( \frac{2a-3}{6} \right) = 2 \times (2a-3)\]
• \[12 \times \left( \frac{3a-2}{4} \right) = 3 \times (3a-2)\]
• \[12 \times \left( \frac{5a+6}{12} \right) = 1 \times (5a+6)\]

This multiplication clears all fractions, leading to simpler terms that are easier to work with." "Fraction elimination is beneficial as it reduces the complexity of working with fractions, enabling a smoother path to solving the equation.

Combining Like Terms

After eliminating fractions, the next step involves Combining Like Terms, which helps to further simplify the equation. Like terms are terms that contain the same variables raised to the same power.
In the equation:

• Identify and group terms with the same variable: \(4a + 9a + 5a\) and constants \(-6 - 6 + 6\).
• Combine these terms by summing or subtracting their coefficients: \((4a + 9a + 5a) = 18a\) and \((-6 - 6 + 6) = -6\).

By combining like terms, the equation becomes more compact and easier to solve. It's a crucial step in the process, as it prepares the equation for final solving steps where the variable will be isolated.

Isolating the Variable

Isolating the Variable is the final step toward finding the solution to your equation. The goal in this step is to get the variable (in this case, \(a\)) on one side of the equation by itself.
Here’s how it’s done:

• First, eliminate any constants from the side with the variable by performing the inverse operation. Add 6 to both sides: \[18a - 6 + 6 = 48 + 6\] which simplifies to \[18a = 54\].
• Next, isolate \(a\) by dividing both sides of the equation by 18, which gets you: \[a = \frac{54}{18}\].
• Simplify the fraction to find \(a = 3\).

Isolating the variable requires careful arithmetic but is straightforward once the equation is fully simplified. It concludes the process of solving linear equations, allowing you to express the variable explicitly.