Problem 56 - Elementary Algebra

Question

Add or subtract as indicated and express your answers in simplest form. (Objective 3) $$\frac{5 x}{2}-\frac{3 x}{4}-\frac{7 x}{6}$$

Step-by-Step Solution

Answer

The answer is \( \frac{7x}{12} \).

Step 1: Identify the Common Denominator

The fractions have denominators 2, 4, and 6. To combine these fractions, we need to find a common denominator that all three share. The least common multiple (LCM) of 2, 4, and 6 is 12.

Step 2: Rewrite Fractions with Common Denominator

Convert each fraction to have the common denominator of 12. This involves adjusting both the numerator and the denominator. \[ \frac{5x}{2} = \frac{5x \times 6}{2 \times 6} = \frac{30x}{12} \] \[ \frac{3x}{4} = \frac{3x \times 3}{4 \times 3} = \frac{9x}{12} \] \[ \frac{7x}{6} = \frac{7x \times 2}{6 \times 2} = \frac{14x}{12} \]

Step 3: Combine the Terms

Since all fractions now share the same denominator, combine them by performing the operations in the numerators:\[ \frac{30x}{12} - \frac{9x}{12} - \frac{14x}{12} \] Combine the terms:\[ \frac{(30x - 9x - 14x)}{12} = \frac{7x}{12} \]

Step 4: Simplify (if possible)

The fraction \( \frac{7x}{12} \) is already in its simplest form as both 7 and 12 have no common factors other than 1.

Key Concepts

Common Denominator

When working with algebraic fractions, finding a common denominator is crucial for adding or subtracting fractions. This process ensures that all fractions involved have the same base number, simplifying the arithmetic. Consider the fractions in our exercise: \(\frac{5x}{2}\), \(\frac{3x}{4}\), and \(\frac{7x}{6}\). Each has different denominators: 2, 4, and 6, respectively. But they must be expressed with a common denominator to perform operations directly and smoothly. A common denominator is essentially a shared multiple of the denominators. Without it, you'd be stuck trying to add or subtract fractions that aren't directly comparable, much like speaking in different languages without translation. To proceed with such fractions, you convert them to have a common denominatorial language, allowing for a straightforward calculation.

Least Common Multiple (LCM)

The concept of the Least Common Multiple (LCM) comes into play when finding a common denominator. The LCM is the smallest number that is a multiple of each of the denominators in the problem. In our exercise, the denominators are 2, 4, and 6. To find the LCM, we look for the smallest number that these denominators all divide into evenly. Listing out multiples of each: 2 produces 2, 4, 6, 8, 10, 12, etc.; multiples of 4 are 4, 8, 12, etc.; and 6 gives 6, 12, 18, etc. Notice that 12 is a number that appears in each list. Thus, 12 is our LCM. By using the LCM, you ensure that the fractions are transformed into an equivalent form with a unified denominator, making addition or subtraction possible. This common denominator allows you to easily work across different fractions, combining their values seamlessly.

Simplest Form

Once all fractions have been adjusted to have a common denominator and their operations are completed, the final result should be expressed in its simplest form. Simplifying a fraction involves reducing it so that the numerator and denominator have no common factors other than 1. In our exercise, after performing the subtraction of the algebraic fractions, we end up with \(\frac{7x}{12}\). Simplification is an essential step; it checks whether the fraction can be broken down further. In this case, the number 7 is a prime number, and 12 is made up of the factors 2 and 3, clearly having no common factors with 7, so \(\frac{7x}{12}\) is already in simplest form. This contributes to a clean, straightforward answer that’s easy to understand and further process in any subsequent calculations.