Simplifying Algebraic Expressions: A Step-by-Step Guide

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Hey guys! Ever find yourself staring at an algebraic expression that looks like a jumbled mess of fractions and variables? Don't worry, you're not alone! Simplifying these expressions is a fundamental skill in mathematics, and once you get the hang of it, it becomes surprisingly straightforward. In this guide, we'll break down the process step-by-step, using the expression 4/7 a + 5/21 a - 2/3 a as our example. So, grab your pencils and let's dive in!

Understanding the Basics: Like Terms

Before we jump into the simplification process, it's crucial to understand the concept of like terms. In algebra, like terms are terms that have the same variable raised to the same power. Think of them as belonging to the same "family." For instance, 3x and 5x are like terms because they both have the variable x raised to the power of 1. Similarly, 2y² and -7y² are like terms because they both have the variable y raised to the power of 2. However, 4x and 4x² are not like terms because the variable x is raised to different powers.

The reason why identifying like terms is so important is that we can only combine like terms through addition and subtraction. It's like trying to add apples and oranges – you can't directly add them, but you can add apples to apples and oranges to oranges. In our expression, 4/7 a, 5/21 a, and -2/3 a are all like terms because they all contain the variable a raised to the power of 1. This means we can combine them to simplify the expression.

When we talk about simplifying algebraic expressions, it's all about making them easier to understand and work with. This often involves reducing the number of terms and making the coefficients (the numbers in front of the variables) as simple as possible. By combining like terms, we're essentially tidying up the expression and presenting it in its most concise form. So, with our understanding of like terms in place, let's move on to the next step: finding a common denominator.

Finding the Common Denominator

Okay, so we've established that we can combine the terms in our expression 4/7 a + 5/21 a - 2/3 a because they are like terms. But, we can't directly add or subtract fractions unless they have the same denominator (the bottom number in a fraction). Think of it like trying to compare slices of pizza – if one pizza is cut into 7 slices and another is cut into 21 slices, it's hard to immediately tell which slice is bigger. We need a common "slice size" to make a fair comparison.

To find the common denominator, we need to determine the least common multiple (LCM) of the denominators 7, 21, and 3. The LCM is the smallest number that all three denominators divide into evenly. Here's how we can find it:

  1. List the multiples of each denominator:
    • Multiples of 7: 7, 14, 21, 28, 35...
    • Multiples of 21: 21, 42, 63...
    • Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24...
  2. Identify the smallest multiple that appears in all three lists: In this case, the smallest common multiple is 21.

Therefore, the least common denominator (LCD) for our fractions is 21. This means we need to rewrite each fraction with a denominator of 21. To do this, we'll multiply the numerator and denominator of each fraction by a suitable factor.

For the first term, 4/7 a, we need to multiply the denominator 7 by 3 to get 21. So, we multiply both the numerator and the denominator by 3:

(4/7) * (3/3) = 12/21

The second term, 5/21 a, already has a denominator of 21, so we don't need to change it.

For the third term, -2/3 a, we need to multiply the denominator 3 by 7 to get 21. So, we multiply both the numerator and the denominator by 7:

(-2/3) * (7/7) = -14/21

Now, our expression looks like this: 12/21 a + 5/21 a - 14/21 a. See how much easier it is to work with now that all the fractions have the same denominator? We're one step closer to simplifying the entire expression!

Combining the Fractions

Alright, guys, we've done the hard work of finding the common denominator, and now we're ready for the fun part: combining the fractions! Remember, now that all our fractions have the same denominator (21), we can simply add and subtract the numerators (the top numbers) while keeping the denominator the same. It's like adding slices of the same-sized pizza – you just count up the number of slices.

Our expression is currently: 12/21 a + 5/21 a - 14/21 a.

To combine the fractions, we'll perform the operations on the numerators:

12 + 5 - 14 = 3

So, when we combine the numerators, we get 3. Now, we write this over our common denominator of 21, and don't forget to include the variable a:

3/21 a

We've successfully combined the fractions! The expression is now simplified to 3/21 a. But, hold on, we're not quite done yet. There's one more step we can take to make our expression even simpler.

Simplifying the Fraction

Great job, guys! We've combined the like terms and arrived at the expression 3/21 a. But, like a diamond in the rough, this fraction has the potential to shine even brighter. What I mean is, we can simplify the fraction 3/21 itself.

Simplifying a fraction means reducing it to its lowest terms. We do this by finding the greatest common factor (GCF) of the numerator and the denominator and then dividing both by that factor. The GCF is the largest number that divides evenly into both the numerator and the denominator. Think of it as finding the biggest "chunk" we can divide both numbers into without leaving any remainders.

In our case, the numerator is 3 and the denominator is 21. Let's find their GCF:

  • Factors of 3: 1, 3
  • Factors of 21: 1, 3, 7, 21

The greatest common factor of 3 and 21 is 3. This means we can divide both the numerator and the denominator by 3:

(3 ÷ 3) / (21 ÷ 3) = 1/7

So, the simplified fraction is 1/7. Now, we replace 3/21 with its simplified form in our expression:

1/7 a

And there you have it! We've successfully simplified the expression 4/7 a + 5/21 a - 2/3 a all the way down to 1/7 a. Pat yourselves on the back, you've conquered a challenging algebraic expression!

Final Result

After following all the steps, we have successfully simplified the expression:

4/7 a + 5/21 a - 2/3 a = 1/7 a

Key Takeaways

Simplifying algebraic expressions might seem daunting at first, but by breaking it down into smaller, manageable steps, it becomes much easier. Here are the key takeaways from this guide:

  1. Identify Like Terms: Only terms with the same variable raised to the same power can be combined.
  2. Find the Common Denominator: When adding or subtracting fractions, they must have a common denominator.
  3. Combine the Numerators: Once you have a common denominator, add or subtract the numerators.
  4. Simplify the Fraction: Reduce the fraction to its lowest terms by dividing the numerator and denominator by their greatest common factor.

By mastering these steps, you'll be well-equipped to tackle a wide range of algebraic expressions. Keep practicing, and you'll become a simplification pro in no time! Remember, math is like a puzzle – the more you practice, the better you get at solving it. So, keep those pencils sharp and keep exploring the wonderful world of algebra!

I hope this guide was helpful in understanding how to simplify algebraic expressions. If you have any questions or want to explore more examples, feel free to ask! Keep up the great work, guys!