Simplify (x+9)/(3y) + (2x)/(3y): A Step-by-Step Guide

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Simplify: (x+9)/(3y) + (2x)/(3y)

Hey guys! Let's break down this math problem together. We've got to simplify the expression (x+9)/(3y) + (2x)/(3y). Don't worry, it's easier than it looks! We're going to walk through each step so you can totally nail it.

Understanding the Problem

Before we dive in, let's make sure we understand what we're looking at. We have two fractions: (x+9)/(3y) and (2x)/(3y). Notice anything special? They both have the same denominator, which is 3y. This is great news because it means we can combine them easily. When fractions have a common denominator, we can simply add their numerators and keep the denominator the same. This is a fundamental rule in fraction arithmetic and is super useful in simplifying algebraic expressions.

When dealing with algebraic expressions, it’s important to pay attention to the details. Make sure you correctly identify the numerators and denominators. The numerator is the expression on top of the fraction bar, and the denominator is the expression below it. In our case, the numerators are (x+9) and (2x), and the denominator is 3y for both fractions. Once you have identified these components correctly, you are ready to combine the fractions.

Also, remember that x and y are variables, which means they represent unknown values. This is why we need to simplify the expression as much as possible, so that it becomes easier to work with if we later need to substitute specific values for x and y. Simplifying expressions is a core skill in algebra, and mastering it will help you tackle more complex problems later on.

Step-by-Step Solution

Okay, let's get to work. Since the denominators are the same, we can add the numerators directly:

(x+9)/(3y) + (2x)/(3y) = (x + 9 + 2x) / (3y)

Now, we need to combine like terms in the numerator. We have x and 2x, which are like terms because they both contain the variable x. Adding them together gives us:

x + 2x = 3x

So, our expression becomes:

(3x + 9) / (3y)

Next, we can factor out a common factor from the numerator. Notice that both 3x and 9 are divisible by 3. Factoring out a 3 gives us:

3(x + 3) / (3y)

Now, we can simplify the fraction by canceling out the common factor of 3 in the numerator and the denominator:

(3(x + 3)) / (3y) = (x + 3) / y

And that's it! We've simplified the expression to (x + 3) / y.

Detailed Breakdown of Each Step

  1. Identify the Common Denominator:

The first step in adding fractions is to ensure they have a common denominator. In this case, both fractions have the denominator 3y. This simplifies the process significantly, as we don't need to find a common denominator—it's already provided.

  1. Combine the Numerators:

Since the denominators are the same, we can add the numerators directly:

(x + 9) + (2x) = x + 9 + 2x

This step involves adding the expressions in the numerators while keeping the denominator constant.

  1. Combine Like Terms:

In the numerator x + 9 + 2x, we can combine the like terms x and 2x. Like terms are terms that have the same variable raised to the same power. In this case, both terms have x raised to the power of 1.

x + 2x = 3x

So the numerator becomes 3x + 9.

  1. Factor Out Common Factors:

Now we have the expression (3x + 9) / (3y). We can factor out the greatest common factor (GCF) from the numerator. The GCF of 3x and 9 is 3.

Factoring out 3 from the numerator gives us:

3(x + 3)

So the expression becomes (3(x + 3)) / (3y).

  1. Simplify the Fraction:

Finally, we can simplify the fraction by canceling out the common factor of 3 in the numerator and the denominator.

(3(x + 3)) / (3y) = (x + 3) / y

This step involves dividing both the numerator and the denominator by the common factor, which simplifies the fraction to its simplest form.

Why This Works

The reason we can add fractions this way is based on the fundamental properties of fractions. When fractions have a common denominator, it means they are divided into the same number of equal parts. So, adding the numerators is like adding the number of parts we have.

Factoring and canceling common factors is also based on the principle of equivalent fractions. When we divide both the numerator and the denominator by the same number, we are essentially multiplying the fraction by 1, which doesn't change its value but simplifies its form.

Choosing the Correct Answer

Looking at the options given:

A) (x+3)/(2y) B) (x+9)/y C) (x+3)/y D) (x+9)/(3y)

The correct answer is C) (x+3)/y, which matches our simplified expression.

Common Mistakes to Avoid

  • Forgetting to Combine Like Terms: Make sure you combine all like terms in the numerator before factoring. This will simplify the expression and reduce the chances of making errors.
  • Incorrectly Factoring: Ensure you factor out the greatest common factor correctly. An incorrect factorization can lead to an incorrect simplification.
  • Canceling Terms Instead of Factors: You can only cancel common factors, not common terms. For example, you cannot cancel the 3 in (3x + 9) / (3y) directly without factoring first.
  • Skipping Steps: It’s tempting to skip steps to save time, but this can lead to errors. Write out each step clearly to minimize mistakes.

Practice Problems

Want to get even better? Try these practice problems:

  1. Simplify: (2a + 5)/(4b) + (a - 1)/(4b)
  2. Simplify: (3x + 2)/(5y) + (x + 8)/(5y)
  3. Simplify: (4p + 7)/(6q) + (2p - 1)/(6q)

Work through these problems step-by-step, and you'll become a pro at simplifying algebraic fractions in no time!

Conclusion

Simplifying (x+9)/(3y) + (2x)/(3y) involves combining the numerators over a common denominator, combining like terms, factoring, and canceling common factors. By following these steps carefully, you can simplify the expression to (x + 3) / y. Keep practicing, and you'll master these types of problems in no time! You got this!