Problem 52 - Elementary Algebra

Question

For the following problems, replace \(N\) with the proper quantity. $$ \frac{y-3}{y+3}=\frac{N}{y^{2}-9} $$

Step-by-Step Solution

Answer

Answer: N = y - 3

Step 1: Factor the denominator on the right side

We first want to factor the denominator on the right side of the equation, \(y^2 - 9\), which is a difference of squares. This can be factored as \((y - 3)(y + 3)\). So the equation becomes: $$ \frac{y-3}{y+3}=\frac{N}{(y-3)(y+3)} $$

Step 2: Compare the numerators

Now that we have factored the denominator on the right side of the equation, we can compare the numerators of both fractions. We see that the numerators must be equal for the two fractions to be equivalent, that is: $$ y - 3 = N $$

Step 3: Replace N with the proper quantity

Now that we have determined that \(N = y - 3\), we can replace \(N\) with this quantity in the original equation. Doing so, we get: $$ \frac{y-3}{y+3}=\frac{y-3}{(y-3)(y+3)} $$

Key Concepts

Factoring Polynomials

Factoring polynomials is an essential technique in algebra that involves rewriting a polynomial as a product of its simpler factors. This process is helpful because it can simplify complex equations and expressions. When you factor a polynomial, you are essentially reversing the process of expansion. For instance, if you have expanded to obtain a polynomial, factoring allows you to find the original expressions that were multiplied together.

To successfully factor a polynomial, you should look for:

• Common factors: Numbers or variables that appear in all the terms of the polynomial.
• Special patterns: Recognize patterns such as the difference of squares or perfect square trinomials.
• Grouping terms: Often useful when a polynomial has four or more terms. Group terms to help find common factors.

In the given exercise, the expression \( y^2 - 9 \) is first identified as a difference of squares, and is then factored into \( (y - 3)(y + 3) \). Thus, understanding how to factor is critical in transforming and simplifying polynomial equations.

Equivalent Fractions

Equivalent fractions are fractions that have the same value, even though they may look different at first glance. To determine if two fractions are equivalent, their cross-products should be equal. This means that multiplying the numerator of one fraction by the denominator of the other should yield the same result as the cross multiplication in the other direction.

In the initial exercise, the task is to ensure the fractions on both sides of the equation are equivalent. By using the steps in factoring, we equate the original fraction to the transformed one. Here’s a simple breakdown:

• The fraction \( \frac{y-3}{y+3} \) is the original expression.
• The objective is to find \( N \) such that \( \frac{N}{y^2 - 9} \) remains unchanged in its value from the original.

Through simplification and comparison of numerators, it is concluded that \( N = y - 3 \), thereby maintaining the equivalence of the fractions.

Difference of Squares

The difference of squares is a specific pattern in algebra where you have an expression in the form \( a^2 - b^2 \). This pattern can be transformed using the identity formula \( a^2 - b^2 = (a - b)(a + b) \). This is a commonly used technique since recognizing this form allows you to quickly factor expressions.

For the problem at hand, the expression \( y^2 - 9 \) is a classic example of the difference of squares:

• It can be rewritten as \( y^2 - 3^2 \).
• Applying the identity formula gives \( (y - 3)(y + 3) \).

Understanding the difference of squares is crucial since it not only simplifies expressions but also broadens the range of algebraic manipulations one can perform. Keeping in mind these identities helps in recognizing and solving polynomial equations efficiently.