Problem 33 - Algebra and Trigonometry

Question

add or subtract as indicated. $$ \frac{4 x+1}{6 x+5}+\frac{8 x+9}{6 x+5} $$

Step-by-Step Solution

Answer

The sum of the two fractions is \( \frac{12x+10}{6x+5} \)

Step 1: Identify Common Denominator

Identify that both fractions have the '6x+5' as a denominator. Therefore, there is no need to find a common denominator.

Step 2: Add Numerators

Add together the numerators of the two fractions: \( (4x+1) + (8x + 9) \)

Step 3: Simplify Numerators

Combine like terms in the numerator to simplify the expression: \(4x + 8x + 1 + 9 \).

Step 4: Write Final Result

Write down the simplified fraction.

Key Concepts

Rational Expressions

A rational expression is essentially a fraction where both the numerator and the denominator are polynomials. In our example, \( \frac{4x+1}{6x+5} \) and \( \frac{8x+9}{6x+5} \) are rational expressions. Understanding rational expressions is crucial in algebra because it allows us to perform operations like addition, subtraction, multiplication, and division with polynomial fractions.

Key aspects to note about rational expressions include:

• The denominator cannot be zero, as this would make the expression undefined.
• Rational expressions can often be simplified by factoring and reducing common factors in the numerator and denominator.

Complex problems may require more steps, but applying the basic rules remains the same. Once you're comfortable recognizing and manipulating these expressions, algebra feels a lot more approachable.

Adding Fractions

Adding fractions involves a few standard steps and rules. Just like regular fractions, you need to ensure that the fractions have a common denominator before you add their numerators.

Let's take this step by step with our given problem. In this case, the fractions \( \frac{4x+1}{6x+5} \) and \( \frac{8x+9}{6x+5} \) already have the same denominator. This is a wonderful shortcut because it eliminates the step of finding a common denominator. Next:

• Add the numerators: \((4x+1) + (8x+9)\).
• Keep the denominator the same: \(6x+5\).

This set up to solve the problem accurately follows the basic principle of adding fractions.

Common Denominator

The common denominator is a fundamental concept when adding or subtracting fractions. It refers to a shared denominator among multiple fractions that allow their numerators to be combined.

In the problem we're addressing, the common denominator is \(6x+5\). Because both fractions in the problem already use \(6x+5\) as their denominator, we don't need additional calculations to create a common denominator. This saves us time and directly leads us to adding the numerators. Efficiently recognizing a shared denominator is a skill that speeds up solving these kinds of algebra problems, making the process much less daunting.

Simplifying Expressions

Simplifying expressions is often the final step in handling algebraic problems like the one given. It means reducing the problem to its simplest form, making it easier to understand and work with.

After you've added the numerators from the initial problem, you get \((4x + 8x + 1 + 9)\), which you can simplify to \(12x + 10\). So, the fraction becomes:

• \( \frac{12x+10}{6x+5} \).

Simplifying involves combining like terms and, if possible, reducing the expression further by factoring. In simpler problems, you might find common factors in the numerator and denominator to cancel out.

The goal is always to express the answer as clearly and concisely as possible. This clarity makes algebra not only simpler but also more rewarding as you solve complex expressions with greater ease.