Problem 30 - Elementary Algebra

Question

For the following problems, add or subtract the rational expressions. $$ \frac{8 p}{11 q}-\frac{3 p}{11 q} $$

Step-by-Step Solution

Answer

Question: Add or subtract the given rational expressions: $$\frac{8p}{11q} - \frac{3p}{11q}$$. Answer: $$\frac{5p}{11q}$$.

Step 1: Identify the common denominator

In this case, the common denominator is already present in both rational expressions, which is \(11q\).

Step 2: Combine the numerators

Now that we have the common denominator, we can go ahead and combine the numerators: $$\frac{8p}{11q} - \frac{3p}{11q}$$. Since they have the same denominator, you simply subtract the numerators (the top parts): $$\frac{8p - 3p}{11q}$$

Step 3: Simplify the expression

Now, simply perform the subtraction operation in the numerator: $$\frac{5p}{11q}$$ So, the final answer is: $$\frac{5p}{11q}$$.

Key Concepts

Common Denominator

Understanding the concept of a common denominator is crucial when it comes to adding or subtracting rational expressions. Rational expressions are fractions that have polynomials in their numerators and denominators. To combine two or more rational expressions, they must have the same denominators, referred to as the common denominator.

Think of the denominator as the shared ground on which the fractions must stand before you can put them together. In the provided exercise, the common denominator for the two expressions \( \frac{8p}{11q} \) and \( \frac{3p}{11q} \) is \( 11q \). Having the same denominator means you can directly combine the numerators without altering the denominators.

To find a common denominatorfor different denominators, you would typically look for the least common multiple (LCM) of the denominators. In situations where the denominators are already the same, as in our exercise, you can proceed to adding or subtracting the numerators directly.

Simplify Rational Expressions

When we simplify rational expressions, we're making them easier to understand and sometimes easier to work with in future algebraic operations. Simplifying involves reducing fractions to their lowest terms, factoring polynomials, and cancelling common factors between the numerator and the denominator.

For instance, once you combine the numerators in the exercise, you end up with \( \frac{8p - 3p}{11q} \). The subtraction in the numerator yields \( \frac{5p}{11q} \), which is already in its simplest form. However, if further simplification were possible, you would look for any common factors in the numerator and denominator that can be divided out.

Simplifying doesn't change the value of the expression; rather, it streamlines it for more straightforward usage.

Rational Expressions Arithmetic

Rational expressions arithmetic behaves similarly to arithmetic with simple fractions. It includes addition, subtraction, multiplication, and division of rational expressions. For addition and subtraction, which we're focusing on, remember this basic rule: only like terms with a common denominator can be combined.

To add or subtract rational expressions:

• Find the common denominator and rewrite each fraction as needed.
• Combine the numerators, making sure to keep track of the signs.
• Simplify the resulting expression, if possible.

Just as the provided exercise illustrates, the numerators of the fractions are subtracted because they share a common denominator. Rational expression arithmetic is all about careful bookkeeping and following the rules for combining terms.

Algebraic Fractions

Algebraic fractions are just like numeric fractions but with variables included in the numerators or denominators. They're also known as rational expressions. Techniques used for numeric fractions, like finding a common denominator, can be applied to algebraic fractions as well.

Working with algebraic fractions often involves simplifying, adding, subtracting, multiplying, and dividing, as you would with numeric fractions. The key difference is keeping track of the variables and applying the rules of algebra, like combining like terms and factoring.

In our exercise, the algebraic fractions \( \frac{8p}{11q} \) and \( \frac{3p}{11q} \) are simplified by subtracting the numerators. The process shows that algebraic fractions can be straightforward when we apply basic fraction rules, with the extra layer of considering variables and polynomial expressions.