Problem 52 - Algebra A Combined Function

Question

Simplify each expression. $$ \frac{49-y^{2}}{y-7} $$

Step-by-Step Solution

Answer

The expression simplifies to \( -(y+7) \).

Step 1: Identify the structure

Notice that the expression \( 49 - y^2 \) resembles a difference of squares, which can be rewritten using the formula \( a^2 - b^2 = (a-b)(a+b) \). Here, \( a^2 = 49 \) and \( b^2 = y^2 \), implying \( a = 7 \) and \( b = y \).

Step 2: Rewrite the numerator

Using the difference of squares formula, we can rewrite \( 49 - y^2 \) as \((7-y)(7+y)\). So we have:\[ \frac{49-y^2}{y-7} = \frac{(7-y)(7+y)}{y-7} \]

Step 3: Simplify the expression

Notice that \((7-y)\) is the negative of \( (y-7) \). Specifically, \( 7-y = -(y-7) \). Therefore:\[ \frac{(7-y)(7+y)}{y-7} = \frac{-(y-7)(7+y)}{y-7} \]

Step 4: Cancel the common factors

Now, we can cancel \( (y-7) \) from the numerator and the denominator, leaving us with:\[ -(7+y) \]

Step 5: Simplify the remaining expression

Distribute the negative sign to get the final simplified expression:\[ -7 - y = -(7+y) \]

Key Concepts

Difference of Squares

The difference of squares is an important concept in algebra, especially when it comes to simplifying expressions. This technique relies on a special formula:

• \( a^2 - b^2 = (a-b)(a+b) \)

The formula tells us that the difference between two squares can be factored into two binomials: one with a subtraction and one with an addition.
In the given exercise, we are dealing with the expression \( 49 - y^2 \). At first glance, it might seem complex, but it matches the difference of squares pattern perfectly:

• \( 49 \) is actually \( 7^2 \)
• \( y^2 \) stays as it is

Thus, you can rewrite \( 49 - y^2 \) as \((7-y)(7+y)\), breaking down the problem into more manageable parts.

Factoring

Factoring is a crucial step in simplifying algebraic expressions, especially when dealing with quartic or polynomial equations. Factoring involves breaking down a complex expression into simpler "factors" that can be multiplied together to get the original expression.
In our exercise, after identifying the difference of squares, we applied the formula to factor \( 49-y^2 \) into \((7-y)(7+y)\). This transforms the expression into a multiplication of two simpler binomials, a vital strategy for further simplification.
The act of factoring reduces the problem into components that are often easier to work with, as seen in further steps of simplifying the original expression. Recognizing when and how to factor expressions forms the backbone for many algebraic solutions.

Canceling Common Factors

Canceling common factors is a simplification technique in which you "cross out" identical terms present in both the numerator and the denominator of a fraction. This helps reduce complex expressions to much simpler forms.
In our example, after factoring the numerator as \((7-y)(7+y)\), we noticed that \((7-y)\) and \((y-7)\) are opposites. Recognizing the relationship \(7-y = -(y-7)\) allowed us to write:

• \(\frac{(7-y)(7+y)}{y-7} = \frac{-(y-7)(7+y)}{y-7} \)

This setup directly exposes a common factor of \((y-7)\) in the numerator and denominator. Cancelation results in a simpler expression: \(-(7+y)\). It's crucial to always check for such common factors to bring simplicity into algebraic expressions.

Algebraic Expressions

Understanding algebraic expressions is fundamental when solving problems related to mathematical simplification, like the one in the exercise. Algebraic expressions consist of variables, constants, and mathematical operations.
The exercise at hand includes the expression \(\frac{49-y^2}{y-7}\), which is a rational algebraic expression due to the presence of variables both in its numerator and denominator.
Such expressions can often appear daunting, but with concepts like factoring, recognizing difference of squares, and canceling, simplification becomes manageable. Mastery of simplifying algebraic expressions illuminates broader mathematical topics, as you use foundational algebraic principles to handle more sophisticated and practical problems in the future.