Combining Like Terms In Algebraic Expressions

Hey guys! Today, we're diving into the world of algebraic expressions and learning how to simplify them by combining like terms. It's like sorting your socks, but with variables and numbers. Let's get started!

What are Like Terms?

Before we jump into the expressions, let's define what like terms actually are. Like terms are terms that have the same variable(s) raised to the same power. The coefficients (the numbers in front of the variables) can be different, but the variable part must be identical. For example, 3x² and -5x² are like terms because they both have x². However, 3x² and 3x are not like terms because one has x² and the other has x.

Why Combine Like Terms?

Combining like terms simplifies algebraic expressions, making them easier to understand and work with. It's like decluttering your room; a tidy expression is much more manageable! By combining like terms, you reduce the number of terms in the expression, which can help in solving equations or further simplifying more complex problems. Think of it as a fundamental step in algebraic manipulation – a skill you'll use throughout your math journey.

Combining like terms also helps in identifying patterns and relationships within the expression. When you group similar terms together, you can easily see which variables have the most significant impact and how they interact with each other. This can be especially useful in real-world applications where algebraic expressions model various phenomena. For instance, in physics, you might use algebraic expressions to represent the motion of an object, and combining like terms can simplify the equations, making it easier to predict the object's behavior. Similarly, in economics, combining like terms in cost and revenue equations can help businesses understand their profitability and make informed decisions. So, mastering the art of combining like terms is not just about simplifying expressions on paper; it's about gaining a deeper understanding of the underlying relationships and patterns in the world around us.

How to Combine Like Terms

Combining like terms involves adding or subtracting the coefficients of the like terms while keeping the variable part the same. Here’s the basic process:

1. Identify the like terms: Look for terms with the same variable(s) raised to the same power.
2. Combine the coefficients: Add or subtract the coefficients of the like terms.
3. Keep the variable part: Write the result with the same variable part as the original like terms.

For example, to combine 3x² + 5x², you would add the coefficients (3 + 5 = 8) and keep the variable part (x²), resulting in 8x².

Let's Tackle the Expressions!

Now, let's apply this knowledge to the expressions you provided.

Expression 1: 5x² + 6 - 2x² + 8

• Step 1: Identify Like Terms
  • 5x² and -2x² are like terms.
  • 6 and 8 are like terms (they are constants).
• Step 2: Combine Like Terms
  • Combine 5x² and -2x²: 5x² - 2x² = 3x²
  • Combine 6 and 8: 6 + 8 = 14
• Step 3: Write the Simplified Expression
  • The simplified expression is 3x² + 14.

So, the like terms are 5x² and -2x², and 6 and 8. The simplified expression is 3x² + 14.

Expression 2: 2/5 ab + ab - 1 + 3b - ab/5

• Step 1: Identify Like Terms
  • 2/5 ab, ab, and -ab/5 are like terms.
  • -1 is a constant.
  • 3b is a term with the variable b.
• Step 2: Combine Like Terms
  • Combine 2/5 ab, ab, and -ab/5: 2/5 ab + ab - ab/5 = 2/5 ab + 5/5 ab - 1/5 ab = (2/5 + 5/5 - 1/5)ab = 6/5 ab
  • The term -1 remains as is.
  • The term 3b remains as is.
• Step 3: Write the Simplified Expression
  • The simplified expression is 6/5 ab + 3b - 1.

Therefore, the like terms are 2/5 ab, ab, and -ab/5. The simplified expression is 6/5 ab + 3b - 1.

Expression 3: 17xy - 43x²y - 12 + 31x²y - 10xy

• Step 1: Identify Like Terms
  • 17xy and -10xy are like terms.
  • -43x²y and 31x²y are like terms.
  • -12 is a constant.
• Step 2: Combine Like Terms
  • Combine 17xy and -10xy: 17xy - 10xy = 7xy
  • Combine -43x²y and 31x²y: -43x²y + 31x²y = -12x²y
  • The term -12 remains as is.
• Step 3: Write the Simplified Expression
  • The simplified expression is 7xy - 12x²y - 12.

So, the like terms are 17xy and -10xy, and -43x²y and 31x²y. The simplified expression is 7xy - 12x²y - 12.

Summary Table

Here's a table summarizing our findings:

Tips and Tricks for Combining Like Terms

To become a pro at combining like terms, here are a few extra tips and tricks:

1. Use colors or symbols: When dealing with complex expressions, use different colors or symbols to mark like terms. This visual aid can help you keep track of which terms go together and prevent you from accidentally combining unlike terms.
2. Rearrange the expression: Sometimes, rearranging the expression can make it easier to identify like terms. Group similar terms together, keeping the signs in front of the terms consistent. For example, rewrite 3x + 2y - x + 5y as 3x - x + 2y + 5y.
3. Pay attention to signs: Always be careful with the signs (+ or -) in front of the terms. Make sure to include the sign when combining coefficients. For instance, in the expression 5x - 3x, the coefficients are 5 and -3, so you would calculate 5 - 3 = 2.
4. Factor out common variables: If you have multiple terms with the same variable but different coefficients, you can factor out the common variable to simplify the expression. For example, ax + bx = (a + b)x. This can be particularly useful when dealing with more complex algebraic expressions.
5. Practice regularly: Like any skill, mastering the art of combining like terms requires practice. Work through a variety of examples, starting with simple expressions and gradually moving on to more complex ones. The more you practice, the more comfortable and confident you'll become.

Real-World Applications

Combining like terms isn't just a theoretical exercise; it has practical applications in various real-world scenarios. Here are a couple of examples:

1. Budgeting and Finance: When managing a budget, you might have different categories of expenses, such as groceries, transportation, and entertainment. By combining like terms (i.e., adding up all the expenses in each category), you can get a clear picture of where your money is going and make informed decisions about how to allocate your resources.
2. Inventory Management: In business, inventory management involves tracking the quantity of different products in stock. By combining like terms (i.e., adding up all the units of the same product), you can determine the total quantity of each product available and make decisions about when to reorder.

Conclusion

And there you have it! Combining like terms is a fundamental skill in algebra that simplifies expressions and makes them easier to work with. By identifying like terms, combining their coefficients, and keeping the variable part the same, you can declutter your algebraic expressions and solve problems more efficiently. Keep practicing, and you'll become a pro in no time! Keep up the great work, and I'll see you in the next math adventure!