Problem 34 - Elementary Algebra

Question

For the following problems, add or subtract the rational expressions. $$ \frac{5 a+1}{a+7}+\frac{2 a-6}{a+7} $$

Step-by-Step Solution

Answer

Answer: The sum of the given rational expressions is \(\frac{7a - 5}{a+7}\).

Step 1: Identify the denominators

The denominators for both rational expressions are the same, (a+7).

Step 2: Combine the numerators

Since the denominators are the same, the numerators can be added together: $$ \frac{(5a + 1) + (2a - 6)}{a+7} $$

Step 3: Simplify the numerator

Combine like terms in the numerator to simplify the expression: $$ \frac{(5a + 2a) + (1 - 6)}{a+7} $$

Step 4: Finish simplifying and write the final answer

Simplify the expression and write the final answer: $$ \frac{7a - 5}{a+7} $$ The sum of the given rational expressions is \(\frac{7a - 5}{a+7}\).

Key Concepts

Adding Rational Expressions

When it comes to adding rational expressions, it's quite similar to adding regular fractions. The key difference is that, instead of numerical denominators, we're dealing with polynomials.
To add rational expressions effectively:

• Ensure the expressions share a common denominator. This makes it convenient to combine them by focusing on the numerators alone.
• Once the denominators are the same, add or subtract the numerators as instructed. You perform this operation as you would with regular numbers, lining up any like terms.

In our exercise, because both rational expressions have the same denominator of \(a+7\), we simply add the numerators, which are \(5a+1\) and \(2a-6\). This gives us the new numerator \((5a+1) + (2a-6)\).

Common Denominators

Understanding common denominators is crucial when working with rational expressions. They allow the expressions to be combined easily.
A common denominator is a shared denominator between two or more rational expressions. Think of it as a common platform that binds different expressions together, making computations straightforward.

• For expressions with the same denominator, addition and subtraction become simple, needing focus only on the numerators.
• For different denominators, we need to transform them into equivalent expressions with a common denominator through techniques like finding the least common multiple.

In this specific problem, both fractions are already arranged perfectly with \(a+7\) as the common denominator. This significantly simplifies the process, removing the need for additional steps to reconcile the denominators.

Simplifying Expressions

Simplifying expressions is all about making them neat and concise. When you're done combining rational expressions, the next step is to simplify the result.
This can often involve:

• Combining like terms: As seen in our example, terms like \(5a\) and \(2a\) from the numerators are added to produce \(7a\).
• Basic math operations: Adjust constants by performing regular integer operations, like combining \(1\) and \(-6\) to get \(-5\).
• Checking for further simplification: Occasionally, there's an opportunity to simplify the entire expression by factoring or reducing. In our example, the final expression \(\frac{7a-5}{a+7}\) is already simplified because there are no common factors in the numerator and denominator.

Once simplified, your expression should be neat, showing the simplest form of the combined rational expressions, ready for interpretation or further computation.