Problem 24 - Algebra A Combined Function

Question

Simplify each expression. $$ \frac{y+9}{9+y} $$

Step-by-Step Solution

Answer

The expression simplifies to 1, except when \( y = -9 \).

Step 1: Recognize Identical Terms

First, notice that the expressions in the numerator and the denominator, \( y + 9 \) and \( 9 + y \), can be rearranged. Rewriting them shows that they are actually the same expression due to the commutative property of addition, which states \( a + b = b + a \).

Step 2: Apply the Commutative Property

Rewrite the denominator \( 9 + y \) as \( y + 9 \). Now the expression becomes \( \frac{y+9}{y+9} \).

Step 3: Simplify the Fraction

Since the numerator and the denominator are identical (\( y+9 \) and \( y+9 \)), the expression simplifies to \( 1 \), provided that \( y \) does not equal \( -9 \) to avoid division by zero.

Key Concepts

Commutative Property

The commutative property of addition is a simple yet fundamental principle in algebra. It tells us that the order in which we add numbers does not affect the result. In mathematical terms, this means:

• \( a + b = b + a \)

For example, \( 2 + 3 = 3 + 2 \), and both equal 5. This is important in algebra because it allows us flexibility in how we rearrange and simplify expressions.
In the exercise, we applied the commutative property to the terms \( y + 9 \) and \( 9 + y \). By recognizing that these expressions are equal due to the commutative property, we could rewrite the denominator to match the numerator. This step is crucial for simplification in algebra and demonstrates the power of rearranging terms.

Fraction Simplification

Fraction simplification is the process of reducing a fraction to its simplest form. This occurs when the numerator and denominator share the same factor, allowing us to divide them, leaving 1 as the result. Here's an easy breakdown:
When you have a fraction like \( \frac{a}{a} \), it simplifies to 1, assuming \( a eq 0 \).
In our problem, after applying the commutative property, both the numerator and denominator were \( y+9 \). Therefore, the entire fraction becomes:

• \( \frac{y+9}{y+9} = 1 \)

This eliminates any complexity in the fraction, simplifying it completely. Fraction simplification is essential in solving algebraic expressions, ensuring expressions are reduced to their most concise and understandable form.

Avoiding Division by Zero

Division by zero is undefined in mathematics because it does not produce a finite result. In our exercise, we reach a simplified fraction, \( \frac{y+9}{y+9} \), which is only valid under the condition that \( y eq -9 \).
Why is this the case? If we substitute \( y = -9 \), we end up with \( \frac{0}{0} \), which is undefined. To avoid division by zero:

• Always identify values that make the denominator zero.
• Exclude those values from the solution set.

By acknowledging this condition, we ensure that our solutions are mathematically sound and free from errors. It is always crucial to check for division by zero when simplifying fractions.