Problem 48 - Introductory Algebra for College Students

Question

denominators are opposites, or additive inverses. Add or subtract as indicated. Simplify the result, if possible. $$\frac{y-7}{y^{2}-16}+\frac{7-y}{16-y^{2}}$$

Step-by-Step Solution

Answer

The result of the expression is 0.

Step 1: Identifying The Denominators

The denominators of the rational expressions, i.e. \(y^{2}-16\) and \(16-y^{2}\) are additive inverses of each other. Thus, these can be rewritten as \(-(-y^{2}+16)\) = \(y^{2}-16\).

Step 2: Combining The Numerators

Since we now have identical denominators, our fraction simplifies to \((y-7)+(7-y)\) divided by their common denominator \(y^{2}-16\). This is equivalent to \(y-7+7-y\)/\(y^{2}-16\).

Step 3: Simplifying The Numerator

By combining the expressions in the numerator, the y's cancel out and -7 + 7 equals 0. So our expression becomes \(0/(y^2-16)\).

Step 4: Fully Simplifying

0 divided by any non-zero number is 0, therefore the final result of the expression is 0.

Key Concepts

Understanding Additive Inverses

In mathematics, the concept of additive inverses is used to describe a pair of numbers that, when added together, result in zero. This principle is crucial when dealing with operations involving rational expressions. For instance, let's consider the exercise provided with denominators being opposites, such as
\frac{y-7}{y^{2}-16}+\frac{7-y}{16-y^{2}}
. Here, the terms \(y^2-16\) and \(16-y^2\) are additive inverses because if we add \(y^2-16\) to \(16-y^2\), the result is zero. Recognizing additive inverses helps in simplifying complex algebraic expressions since it provides a pathway to combine or cancel out terms. It is instrumental in the steps to resolve an equation or simplify an expression to its most basic form.

Combining Like Terms

Combining like terms is a fundamental process in algebra which involves simplifying algebraic expressions by merging terms that have the identical variables raised to the same power. To correctly combine like terms, follow these guidelines: focus on the coefficients (the numerical parts) of the like terms, use the appropriate arithmetic operations (addition or subtraction), and keep the variable part unchanged.
In the given exercise, once we recognized that we have identical denominators, the step was to combine the numerators. This step required us to merge \(y-7\) and \(7-y\), which are like terms because they consist of the same variable, y. By rearranging and combining these terms, we observe that they cancel each other out. Through this straightforward technique, we can often simplify expressions that may seem complicated at the outset.

Working with Algebraic Fractions

Algebraic fractions, also known as rational expressions, are fractions whose numerator and denominator are polynomials. Just like with ordinary fractions, it's possible to add, subtract, multiply, and divide algebraic fractions by following specific algebraic rules. Simplifying algebraic fractions is an essential skill, as it may lead to easier calculations and a clearer understanding of the expression.
In our example, simplifying the algebraic fraction involved reducing it to the form \(0/(y^2-16)\), which may seem unusual. However, recall that any nonzero number divided by itself is 1, and 0 divided by any nonzero number is 0. Therefore, this simplified form is perfectly valid, and it highlights an essential characteristic of algebraic fractions: they abide by the same numerical rules we use for simple arithmetic fractions.