Problem 66 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \frac{8}{4 a+1}+\frac{5}{4 a+1} $$

Step-by-Step Solution

Answer

The simplified expression is: \( \frac{13}{4a+1} \)

Step 1: Analyze the fractions and their denominators

The given expression consists of two fractions, \(\frac{8}{4a+1}\) and \(\frac{5}{4a+1}\). These two fractions have the same denominator \(4a+1\).

Step 2: Add the Fractions

For fractions that have the same denominator, we can simply add the numerators and keep the denominator the same. Therefore \(\frac{8}{4a+1} + \frac{5}{4a+1}\) becomes \(\frac{8+5}{4a+1}\)

Step 3: Simplify the result

Adding the numerators gives us \(\frac{13}{4a+1}\)

Key Concepts

Simplifying Fractions

When dealing with fractions, simplification is an essential skill that helps in making expressions more manageable. Simplifying a fraction means reducing it to its simplest form where the numerator and denominator have no common factors besides 1. This process makes it easier to understand and use the fraction in calculations.
To simplify a fraction, follow these steps:

• Identify the greatest common factor (GCF) of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCF.

In our exercise, after combining the fractions, we got \(\frac{13}{4a+1} \).Here, 13 and \(4a + 1\) have no common factors, so the expression is already in its simplest form!

Common Denominator

A common denominator is crucial when working with multiple fractions, especially for adding or subtracting them. It refers to a shared denominator that allows for easy combination of such fractions.
Here's why a common denominator is important:

• Enables straightforward addition and subtraction by aligning denominators.
• Eliminates the need for complex calculations with unequal parts.

In the original exercise, both fractions \(\frac{8}{4a+1}\) and \(\frac{5}{4a+1}\) already have the common denominator of \(4a+1\). This makes the process of adding them straightforward.

Adding Fractions

Adding fractions is simple when you have a common denominator. This condition allows you to directly add the numerators while keeping the same denominator.
Here's the process in detail:

• Ensure both fractions share the same denominator.
• Add the numerators together to form a new numerator.
• Keep the original common denominator.

For our expression \(\frac{8}{4a+1} + \frac{5}{4a+1}\):

• Both fractions have the denominator \(4a+1\).
• Add the numerators: 8 and 5, resulting in 13.
• Form the fraction \(\frac{13}{4a+1}\).

With a shared denominator, adding fractions becomes a simple matter of summing up the numerators!