Problem 22 - Intermediate Algebra

Question

Add or subtract, and then simplify, if possible. See Example 1. $$\frac{16 x}{3 z^{2}}-\frac{x}{3 z^{2}}$$

Step-by-Step Solution

Answer

The simplified form is \(\frac{5x}{z^2}\).

Step 1: Identify the Problem

We need to subtract two fractions that have like denominators: \(\frac{16x}{3z^2}\) and \(\frac{x}{3z^2}\). Our goal is to find their difference and simplify the result.

Step 2: Subtract the Numerators

Since the fractions have the same denominator, subtract their numerators directly. The formula for subtraction is \(\frac{a}{c} - \frac{b}{c} = \frac{a-b}{c}\). For our problem, this becomes:\[\frac{16x - x}{3z^2} = \frac{15x}{3z^2}\]

Step 3: Simplify the Fraction

Now, simplify \(\frac{15x}{3z^2}\) by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 15 and 3 is 3.Divide the numerator and the denominator by 3:\[\frac{15x}{3z^2} = \frac{15x \div 3}{3z^2 \div 3} = \frac{5x}{z^2}\]

Step 4: Write the Final Answer

The simplified form of the expression is \(\frac{5x}{z^2}\). Both the numerator and the denominator have no additional common factors, so this is the final answer.

Key Concepts

Subtracting Fractions

When subtracting algebraic fractions, the process is similar to subtracting numerical fractions. The most important aspect in both cases is the denominator. If the fractions already have a common denominator, you can subtract the numerators directly. This simplifies the subtraction process significantly.

Consider a scenario where we have two fractions, \(\frac{a}{c}\) and \(\frac{b}{c}\). Since both have the same denominator, the subtraction becomes simply subtracting their numerators:

• Subtract the numerators: \(a - b\)
• Use the common denominator: \(c\)

So the resultant fraction becomes \(\frac{a - b}{c}\). This method makes subtracting fractions straightforward when the denominators are alike. Remember that you only deal with the numerators, and the denominator remains unchanged.

In our example, we subtracted \(\frac{x}{3z^2}\) from \(\frac{16x}{3z^2}\). Since they both share the denominator \(3z^2\), we just subtracted the numerators to get \(\frac{15x}{3z^2}\). Easy, right?

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. The goal is to have both the numerator and denominator in the smallest numbers possible, that they can be, without changing the value of the fraction.

To simplify, divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD is the largest number that can evenly divide both the numerator and the denominator. Simplifying not only makes calculations easier but also helps in spotting mathematical relationships in algebraic expressions.

• Find the GCD of the numerator and the denominator.
• Divide both by the GCD.

In our specific exercise, after subtracting, we had \(\frac{15x}{3z^2}\). Here, the GCD of 15 and 3 is 3. By dividing both by 3, we simplified the fraction to \(\frac{5x}{z^2}\). Simplifying is the final step in making your answer clean and neat.

Greatest Common Divisor (GCD)

The Greatest Common Divisor, often abbreviated as GCD, plays a crucial role in simplifying fractions. It tells us the largest number that can divide both the numerator and the denominator, helping us reduce fractions to their simplest form.

To find the GCD, you can list out the factors of each number and look for the largest one they share. Alternatively, you can use the Euclidean algorithm, which is a more efficient method for larger numbers.

In the case of our example, we needed to simplify \(\frac{15x}{3z^2}\). We focused only on the numerical parts, 15 and 3.

• Factors of 15: 1, 3, 5, 15
• Factors of 3: 1, 3

The largest number common to both lists is 3, making it the GCD. Thus, we divided the numerator and denominator by 3, greatly simplifying our fraction to \(\frac{5x}{z^2}\). Understanding the GCD makes the process of simplifying fractions smoother and more logical.