Problem 35 - Elementary Algebra

Question

For the following problems, add or subtract the rational expressions. $$ \frac{x+1}{5 x}+\frac{z+3}{5 x} $$

Step-by-Step Solution

Answer

Question: Add the following rational expressions and provide the simplified form: $$ \frac{x+1}{5x} + \frac{z+3}{5x} $$ Answer: The simplified form of the given expression is: $$ \frac{x + z + 4}{5x} $$

Step 1: Identify the common denominator

Since both fractions have the same denominator, \(5x\), the common denominator is simply \(5x\).

Step 2: Combine the numerators

With the common denominator already in place, we can now combine the numerators of both fractions by adding them together: $$ \frac{x+1}{5x} + \frac{z+3}{5x} = \frac{(x+1) + (z+3)}{5x}. $$

Step 3: Simplify the numerator

Now, simplify the combined numerator by performing the addition operation: $$ \frac{(x+1) + (z+3)}{5x} = \frac{x + 1 + z + 3}{5x} = \frac{x + z + 4}{5x}. $$

Step 4: Check for further simplification

The resulting fraction, \(\frac{x + z + 4}{5x}\), cannot be simplified further, as there are no common factors between the numerator and denominator that can be canceled out. Therefore, the final simplified form of the given expression is: $$ \frac{x + z + 4}{5x}. $$

Key Concepts

Common Denominator

Adding or subtracting algebraic fractions requires a common ground, much like finding a universal language for an international meeting. This is where the idea of a common denominator comes into play. Just like having a translator ensures that everyone understands each other, a common denominator allows us to add or subtract fractions with clarity.

A common denominator is essentially the shared multiple of the denominators of two or more fractions. In algebra, we mainly look for the least common denominator (LCD) to make the calculation as simple as possible. It's similar to finding the least disruptive way to ensure that everyone is on the same page during a conversation.

When working with algebraic expressions, if the denominators are already the same, as they are in the given exercise with both denominators being 5x, there's no need to look for an LCD. Just like if everyone in a meeting already speaks English, there’s no need for a translator. Thus, the common denominator in our case is simply \(5x\). This serves as our foundation to unify the numerators in the next step.

Simplify Numerators

With a common denominator established, we can turn our focus to the upper portion of our fractions - the numerators. Think of the numerator as the specific statement someone wants to make; it conveys the unique part of each faction. When simplifying numerators, we are essentially streamlining these statements to make them as direct and simple as possible.

To simplify, we combine the numerators by performing basic algebraic addition or subtraction. For our problem, that means taking \(x+1\) and \(z+3\) and merging them together. The process is akin to summarizing the points of speakers in a meeting: \(x+1\) plus \(z+3\) becomes \(x+z+4\). Simplifying helps us make the overall fraction a clearer and more concise representation of the combined original expressions. The operation is straightforward when the denominators are the same, as no complex manipulation is required - just combine like terms.

Algebraic Fractions

Finally, let's take a step back and look at algebraic fractions as a whole. These are just like the fractions we're used to, but instead of numbers, they contain variables, constants, and sometimes exponents. Their complexity can range from the very simple to the fiendishly complex.

Algebraic fractions follow the same rules for addition, subtraction, multiplication, and division as numerical fractions. But they also open the door to algebraic manipulation, which can be a powerful tool when dealing with unknown values or variables. In this exercise, our algebraic fractions were simple, and the variables did not require any advanced techniques to add.

Once we have our simplified numerator over the common denominator, we must look for any further opportunities to simplify, as is the rule with all fractions. In this case, there are no common factors to cancel, and the expression \(\frac{x+z+4}{5x}\) is already in its simplest form. This final fraction represents the sum of our two original algebraic fractions, elegantly illustrating the concept of adding fractions in the realm of algebra.