Problem 96 - Elementary Algebra

Question

For the following problems, add or subtract the rational expressions. $$ \frac{-a+7}{8-3 a}+\frac{2 a+1}{3 a-8} $$

Step-by-Step Solution

Answer

Question: Add the following rational expressions and simplify the result: \(\frac{-a+7}{8-3a} + \frac{2a+1}{3a-8}\). Answer: The sum of the given rational expressions is \(\frac{-9a^2 + 48a - 54}{(8-3a)(3a-8)}\).

Step 1: Identify the Denominators of the Two Fractions

The denominators of the two given fractions are \((8-3a)\) and \((3a-8)\).

Step 2: Find the Least Common Denominator (LCD)

As the denominators are not primed with respect to one another, i.e., \((3a-8) \neq \pm(8-3a)\), the LCD will be the product of both denominators. Hence, \({\rm LCD} = (8-3a)(3a-8)\).

Step 3: Rewrite Each Fraction with the LCD as the New Denominator

We will now rewrite both fractions with the LCD as the new denominator: $$ \frac{-a+7}{8-3a}\cdot\frac{3a-8}{3a-8}+\frac{2a+1}{3a-8}\cdot\frac{8-3a}{8-3a}. $$ Doing so yields: $$ \frac{(-a+7)(3a-8)}{(8-3a)(3a-8)} + \frac{(2a+1)(8-3a)}{(8-3a)(3a-8)}. $$

Step 4: Add the Numerators Together and Simplify

Let's add the numerators of the two fractions together: $$ \frac{(-a+7)(3a-8)+(2a+1)(8-3a)}{(8-3a)(3a-8)}. $$ Expanding the numerators, we get: $$ \frac{-3a^2 + 24a + 8a - 56 + 16a + 2 - 6a^2 + 3a}{(8-3a)(3a-8)}. $$ Now, we can combine like terms in the numerator: $$ \frac{-9a^2 + 48a - 54}{(8-3a)(3a-8)}. $$ The expression is now simplified, and the final result is: $$ \frac{-9a^2 + 48a - 54}{(8-3a)(3a-8)}. $$

Key Concepts

Rational Expressions

When we talk about rational expressions, we are referring to fractions where the numerator and the denominator are both polynomials. Just like regular fractions, they represent division, but the variables included make them a bit trickier to work with.

In our exercise, the expressions \(\frac{-a+7}{8-3a}\) and \(\frac{2a+1}{3a-8}\) are considered rational because \-a+7\ and \(2a+1\) (the numerators) as well as \(8-3a\) and \(3a-8\) (the denominators) are polynomials. The key to adding or subtracting these expressions, just like with common fractions, lies in finding a common denominator—this is essential for performing the operation correctly.

Least Common Denominator (LCD)

The least common denominator (LCD) refers to the least common multiple of the denominators of two or more fractions or rational expressions. It’s the smallest expression that each of the denominators can divide into without leaving a remainder.

In our example, we can't directly add or subtract \(\frac{-a+7}{8-3a}\) and \(\frac{2a+1}{3a-8}\) because their denominators are different. However, we notice that \(8-3a\) and \(3a-8\) are negatives of each other. By multiplying these two expressions, we get the least common denominator that works for both original expressions. This process involves recognizing patterns, factoring, or sometimes long division, ensuring we simplify the LCD as much as possible. Remember, using the LCD is all about making our rational expressions 'speak the same language' so that they can be easily combined.

Simplifying Algebraic Expressions

The process of simplifying algebraic expressions is to reduce them to their simplest form without changing their value. When we simplify, we combine like terms, which are terms that have the same variables raised to the same power, and we also simplify the coefficients, the numerical factors in the terms.

After finding the LCD and combining the rational expressions in our workout, we reach a point where we have to simplify the complex numerator. To do that, we expand the algebraic expressions, combine like terms, and then reduce the coefficients if possible. Simplification makes our expression neater and often easier to understand or further manipulate. It could also reveal further opportunities to factor and reduce the expression, leading to an even simpler form.