Problem 41 - Elementary Algebra

Question

For the following problems, add or subtract the rational expressions. $$ \frac{3 y+4}{y+8}-\frac{2 y-5}{y+8} $$

Step-by-Step Solution

Answer

$$ \frac{3y + 4}{y + 8} - \frac{2y - 5}{y + 8} $$ Answer: The simplified expression after subtracting the given rational expressions is $$\frac{y+9}{y+8}$$.

Step 1: Identify the denominators

The given rational expressions have the same denominator, which is (y+8).

Step 2: Perform the subtraction in the numerators

Since the denominators are the same, we can perform the subtraction in the numerators: $$ \frac{3 y+4}{y+8} - \frac{2 y-5}{y+8} $$ $$ = \frac{(3 y+4) - (2 y-5)}{y+8} $$

Step 3: Simplify the numerator

Combine the like terms in the numerator: $$ = \frac{3y + 4 - 2y+ 5}{y+8} $$ $$ = \frac{3y - 2y}{y+8} + \frac{4 + 5}{y+8} $$ $$ = \frac{y}{y+8} + \frac{9}{y+8} $$

Step 4: Combine the terms

Now we can combine the terms over the common denominator: $$ = \frac{y + 9}{y+8} $$ The simplified expression after addition or subtraction of the given rational expressions is: $$ \frac{y+9}{y+8} $$

Key Concepts

Rational Expressions

Understanding rational expressions is crucial when working with polynomials and algebra. A rational expression is a fraction that has polynomials in both its numerator and denominator. Much like fractions with numbers, rational expressions can be added, subtracted, multiplied, and divided. Just remember the key rule: The denominator cannot be zero, as division by zero is undefined. Simplifying rational expressions involves factoring polynomials and reducing them to their lowest terms when possible.

Common tasks with rational expressions include finding a common denominator, simplifying complex fractions, and solving equations that contain them. It becomes particularly useful to know how to handle rational expressions when they appear in calculus, such as in integration and limits.

Simplifying Expressions

Simplifying expressions is a foundational skill in algebra. It involves rewriting expressions in a more concise form without changing their value. To simplify an expression, combine like terms and use the distributive property to eliminate parentheses when necessary. It's also important to reduce fractions to lowest terms where possible.

When working with rational expressions, simplification can also mean finding a single expression when given a sum or difference. In the original exercise, 'simplify' refers to combining like terms in the numerator and then bringing together terms over a common denominator to reach the simplest form possible for the rational expression.

Like Terms

Like terms are terms that contain the same variables raised to the same power. For instance, in the expression of the exercise, both terms in the numerator, after expansion \(3y + 4 - 2y + 5\)\), include 'y' terms (\(3y\)\) and (\(2y\)\)) and constant terms (\(4\)\) and (\(5\)\)). They can be combined because they are like terms.

When simplifying expressions, always look for like terms to combine. This simplification, in essence, is just performing basic addition or subtraction on the coefficients while keeping the variable parts intact.

Common Denominator

Adding and subtracting rational expressions require a common denominator, similar to the process with numerical fractions. A common denominator is a shared multiple of the denominators of two or more fractions. With a common denominator, the numerators can be directly added or subtracted while maintaining the shared denominator.

In the given exercise, the common denominator is \(y + 8\)\). Having the same denominator in both rational expressions allows for straightforward subtraction and addition of the numerators. After the operation on the numerators, if the rational expression can be further simplified, such as combining like terms or factoring, those steps should be completed to achieve the simplest form.