Problem 23 - Intermediate Algebra

Question

Add or subtract, and then simplify, if possible. See Example 1. $$\frac{4 y}{y-4}-\frac{16}{y-4}$$

Step-by-Step Solution

Answer

The expression simplifies to 4.

Step 1: Identify Like Denominators

Both fractions \( \frac{4y}{y-4} \) and \( \frac{16}{y-4} \) have the same denominator \( y-4 \). This means that we can add or subtract the fractions directly by working on their numerators.

Step 2: Subtract the Numerators

Since the denominators are the same, subtract the numerators:\[\frac{4y - 16}{y-4}\]

Step 3: Simplify the Numerator

Factor the numerator \( 4y - 16 \). Notice that both terms share a common factor of 4.\[4y - 16 = 4(y - 4)\]

Step 4: Perform Cancellation

The expression now is:\[\frac{4(y-4)}{y-4}\]Since \( y-4 \) is a common factor in both the numerator and denominator, they can be canceled:\[4\]

Step 5: Confirm Simplification

After canceling \( y-4 \) from both the numerator and the denominator, the simplified expression is just 4. This confirms that the original expression simplifies correctly.

Key Concepts

Adding and Subtracting Fractions

Adding and subtracting fractions in algebra works quite similarly to how you'd do so with ordinary numbers. The key is to always look at the denominators of the fractions first. If they are the same, you're in luck! You can simply add or subtract the numerators as you would with normal numbers. In our exercise, both fractions \( \frac{4y}{y-4} \) and \( \frac{16}{y-4} \) have a common denominator, \( y-4 \). This allows the direct subtraction of the numerators, resulting in the expression \( \frac{4y - 16}{y-4} \).

• Ensure denominators are identical.
• Add or subtract the numerators.

If the denominators are not the same, you'll need to find a common denominator before combining the fractions. This is a more involved process, which you may encounter in other problems. It's crucial to get comfortable with these concepts, as they form the foundation for dealing with more complex algebraic fractions.

Simplifying Expressions

Once you've performed the addition or subtraction of fractions, the next step is often to simplify the resulting expression. Simplifying means reducing it to its most basic form without changing its value and helps make the expression easier to handle. After subtracting the fractions in the exercise, we reached the expression \( \frac{4y - 16}{y-4} \). This expression can be simplified by factoring the numerator.

• After addition or subtraction, always check if you can factor.
• Look for common factors.

Simplification often involves finding a greatest common factor, which in this case, is 4 (making \( 4(y - 4) \)). This factored form permits further simplification.

Factorization in Algebra

Factorization involves breaking down expressions into products of simpler expressions, and it is a powerful tool in algebra. In the context of simplifying algebraic fractions, factorization can often allow you to cancel out terms and greatly simplify your expression. In the exercise, the numerator \( 4y - 16 \) was factored into \( 4(y - 4) \). Recognizing that \( y-4 \) was also in the denominator enabled us to cancel the terms, leaving \( 4 \) as the final simplified result.

• Factor to find common components in numerator and denominator.
• Cancel out common factors for simplification.

Mastering factorization will not only help in simplifying expressions but also in solving equations and understanding polynomial functions. Look for patterns and common terms, and practice factorization to strengthen your algebraic skills.