Question

Find the sum or difference. \(\frac{5 x}{x+3}+\frac{15}{x+3}\)

Short answer

The simplified expression is 5.

Step-by-step solution

Identify the Common Denominator

In the equation \(\frac{5x}{x+3} + \frac{15}{x+3}\), both terms share a common denominator of \(x+3\). Thus, these fractions can be easily added.

Combine the Terms

Combine the terms by adding the numerators and keeping the denominator \(x+3\) constant. This will give a new equation: \(\frac{5x+15}{x+3}\).

Simplify the Fraction

The fraction \(\frac{5x+15}{x+3}\) can be simplified by factoring out a '5' from the numerator, yielding \(\frac{5(x+3)}{x+3}\).

Cancel Common Terms

After factoring, we notice that \(x+3\) exists in both the numerator and the denominator. They cancel each other out leaving the final simplified expression, '5'.

Key concepts

Common Denominator

When working with algebraic fractions, a crucial first step is identifying a common denominator. A common denominator allows you to add or subtract fractions with ease. In the given problem, \(\frac{5x}{x+3} + \frac{15}{x+3}\), both fractions naturally share the denominator \(x+3\). This simplifies the process since we don't have to modify any fraction to match another. Here's why a common denominator is essential:

• Ensures the fractions are measured against the same "whole," making addition and subtraction valid.
• Allows the direct addition or subtraction of numerators while keeping the denominator constant.

To find a common denominator, you can:

• Multiply the denominators if they are different. For example, if you have denominators \(a\) and \(b\), the common denominator is \(a \times b\).
• Use the least common multiple (LCM) when possible, especially for numerical denominators.

In this problem, the work is straightforward since the denominators are already the same. This makes it simple to combine the fractions and move to the next step, simplifying.

Simplifying Fractions

Once you have combined fractions with a common denominator, the next crucial step is simplifying. Simplifying fractions makes them easier to work with and often gives a cleaner result. In our exercise, after combining the terms, the fraction becomes \(\frac{5x + 15}{x + 3}\). We can simplify this by factoring the numerator.Here's how simplification works:

• Factor out the greatest common factor (GCF) from the numerator. In \(5x + 15\), both terms are divisible by 5.
• This gives us \(\frac{5(x+3)}{x+3}\).

Once factored, simplification involves canceling out terms that appear in both the numerator and the denominator.This step is significant because:

• It reduces the fraction to its simplest form, often exposing solutions more clearly.
• Ensures that you work with the neatest expression possible, which helps avoid errors in further calculations.

In our example, the expression simplifies to '5' after canceling \(x+3\). Simplification is like tidying up your mathematical solution, making it neat and concise.

Add and Subtract Fractions with Like Denominators

Adding and subtracting fractions becomes straightforward when they have "like" or common denominators. This means that the denominator is the same across the fractions you are working with. In the exercise \(\frac{5x}{x+3} + \frac{15}{x+3}\), since both fractions share the denominator \(x+3\), you can seamlessly add the numerators. Here's a step-by-step method:

• Add or subtract the numerators: simply perform the arithmetic operation on the numerators while keeping the denominator constant.
• The result is a single fraction with the shared denominator. In our case, this yields \(\frac{5x + 15}{x+3}\).

This process is simple because:

• It allows you to focus only on the numerators, making the operation less tedious.
• Ensures that the fractions remain consistent—nothing changes in their relative value or size as measured by the denominator.

Adding or subtracting with like denominators is a fundamental skill, often used as a base for more complex algebraic manipulations.