Problem 69 - Algebra A Combined Function

Question

Write two rational expressions with the same denominator whose sum is \(\frac{5}{3 x-1}\).

Step-by-Step Solution

Answer

\(\frac{2}{3x-1}\) and \(\frac{3}{3x-1}\)

Step 1: Understand the Problem

We need to create two rational expressions with the same denominator whose sum is equal to \(\frac{5}{3x-1}\). This means if we add these two fractions, the numerator should sum to 5 while maintaining the denominator \(3x - 1\).

Step 2: Choose Denominator

Since both rational expressions must have the same denominator, we'll choose \(3x - 1\) for both, as given in the sum \(\frac{5}{3x-1}\).

Step 3: Choose Numerators

We need numerators that sum to 5. Let's choose two simple numbers like 2 and 3, since \(2 + 3 = 5\).

Step 4: Write Expressions

Now, write the two rational expressions with the chosen numerators and denominator: \(\frac{2}{3x-1}\) and \(\frac{3}{3x-1}\).

Step 5: Verify the Sum

Add the expressions to ensure they give the right sum:\[\frac{2}{3x-1} + \frac{3}{3x-1} = \frac{2+3}{3x-1} = \frac{5}{3x-1}\]The sum matches the given expression, so the solution is verified.

Key Concepts

The Importance of a Common Denominator

When adding rational expressions, the denominator plays a key role. The denominator represents the bottom part of a fraction, and to add fractions seamlessly, they must have the same denominator. This is known as having a "common denominator."
In this exercise, we are given fractions with the denominator of \(3x - 1\). By maintaining this common denominator in both fractions, it allows us to easily add the numerators.
Think of the common denominator as a kind of "frame" that holds each fraction in the same space. Without a common denominator, adding fractions would be like trying to add apples and oranges. They simply don’t naturally combine without adjusting the denominator first.
To find a common denominator, you typically look for the least common multiple (LCM) of the denominators, although in this exercise, it was given directly.

Understanding and Choosing Numerators

The numerator, or the top number in a fraction, shows how many parts of the whole are being considered. When dealing with rational expressions, choosing the right numerators is crucial for getting the correct sum.
In our example, we are aiming for a total numerator of 5. Thus, any two numerators that add together to make 5 will work. For simplicity, the numerators 2 and 3 were chosen in this case.
Choosing numerators means deciding on specific parts of the "whole" that you will add together. Unlike the denominator, numerators directly alter the value of the sum of the fractions. Remember, different pairs like 1 and 4, or even 2.5 and 2.5, could work here too. What matters is the final sum, while keeping the denominator constant.

The Art of Addition of Fractions

When adding fractions, the process is much simpler once you have a common denominator. With a common denominator, you only need to add the numerators, keeping the denominator the same. This is exactly what we did in the solution.
To add fractions like \(\frac{2}{3x-1}\) and \(\frac{3}{3x-1}\), you simply add the numerators: \(2 + 3 = 5\). This sum of the numerators over the common denominator gives us \(\frac{5}{3x-1}\), exactly what was required in the solution.
Keep in mind, addition without a common denominator involves finding equivalent fractions that do share one. However, once a common denominator is found or given, it's a straightforward process to add fractions: sum the numerators and keep the denominator constant. This method simplifies what can initially seem a daunting task.