Problem 14 - Algebra 1

Question

Simplify the expression. $$\frac{3 x}{4 x+1}+\frac{5 x}{4 x+1}$$

Step-by-Step Solution

Answer

The simplified expression for \(\frac{3 x}{4 x+1} + \frac{5 x}{4 x+1}\) is \(\frac{8x}{4 x+1}\).

Step 1: Recognize Identical Denominators

Recognize that the denominators in both fractions are the same, which are \(4x + 1\). Therefore, they can be added directly.

Step 2: Add the Numerators

Since the denominators are the same, add the numerators. The numerator of the first fraction is \(3x\) and the numerator of the second fraction is \(5x\). Their sum is \(3x+5x\), which simplifies to \(8x\).

Step 3: Write the Simplified Expression

The sum of the two fractions is then the added numerators over the common denominator, equivalent to \(\frac{8x}{4 x+1}\).

Key Concepts

Identical Denominators in Algebraic Fractions

Grasping the concept of identical denominators is crucial when simplifying algebraic expressions that involve fractions. In mathematics, a denominator is the number or expression written below the line in a fraction, which represents the total number of parts. When combining fractions, if the denominators are the same, we call these identical denominators, and it simplifies the process significantly.

Imagine you have a pizza divided into equal parts. If you and your friend have slices from the same pizza, it's straightforward to figure out how many slices you have together. This is because the size of the slices (denominators) is the same for both of you. In algebraic terms, having identical denominators allows you to focus on combining the numerators directly, without modifying the denominators, just like adding pizza slices. For example, with the expression \(\frac{3x}{4x+1} + \frac{5x}{4x+1}\), the identical denominator is \(4x + 1\). Since they are the same, we can add the numerators without altering the denominators, streamlining the simplification process.

Adding Numerators in Algebraic Fractions

Once you've identified that your algebraic fractions have identical denominators, the next step is to add the numerators. The numerator, which is the expression above the line in a fraction, represents part of the whole. In an algebraic setting, numerators can include variables, constants, or a combination of both. When their denominators match, the numerators can be added as one would add regular numbers.

Consider the analogy of having several bags of marbles with equal numbers of marbles in each. If you only counted the marbles in two bags, you would be adding the numerators (the count of marbles from each bag), not the total capacity of the bags, which is consistent (the denominators). In the exercise \(\frac{3x}{4x+1} + \frac{5x}{4x+1}\), we simply combine the numerators, obtaining \(3x + 5x = 8x\). This combined numerator reflects the sum of individual parts while keeping the common denominator intact.

Understanding Algebraic Fractions

Algebraic fractions may seem daunting at first, but they are just an extension of the fractions we commonly know, incorporating variables like \(x\), \(y\), and others. An algebraic fraction is formed when you have a fraction with a numerator, a denominator, or both containing algebraic expressions.

In daily life, an algebraic fraction can be likened to a recipe that varies depending on the number of people you're cooking for. The ingredients (variables) change proportionally, just like the numerator and denominator in an algebraic fraction. Simplifying these expressions often involves finding common denominators and combining numerators, as we do with regular fractions. Simplification can include adding, subtracting, multiplying, or dividing the fractions, depending on what is required. In our example, the process resulted in the simplified expression \(\frac{8x}{4x+1}\). In essence, understanding algebraic fractions is about recognizing patterns and applying arithmetic operations while keeping an eye on the variables involved.