Combine Like Terms To Create An Equivalent Expression. 1.3b+7.8-3.2b

Combining Like Terms: A Comprehensive Guide to Simplifying Algebraic Expressions

Combining like terms is a fundamental concept in algebra, crucial for simplifying expressions and solving equations. It's a process that allows us to streamline complex mathematical statements into more manageable and understandable forms. This article will delve deep into the process, explaining what like terms are, how to identify them, and how to combine them effectively, using the example expression 1.3b + 7.8 - 3.2b as a running example. We'll also explore various applications and common mistakes to avoid.

Understanding Like Terms

Before diving into combining like terms, it's vital to understand what constitutes a "like term." Like terms are terms that have the same variables raised to the same powers. The coefficients (the numbers in front of the variables) can be different, but the variable part must be identical.

Let's break down the example expression: 1.3b + 7.8 - 3.2b

• 1.3b: This term contains the variable 'b' raised to the power of 1 (remember, if no exponent is written, it's understood to be 1). The coefficient is 1.3.
• 7.8: This term is a constant. It doesn't contain any variables.
• -3.2b: This term also contains the variable 'b' raised to the power of 1. The coefficient is -3.2.

Notice that 1.3b and -3.2b are like terms because they both have the variable 'b' raised to the power of 1. The constant term 7.8 is not a like term to either of the others.

Identifying Like Terms in Complex Expressions

Identifying like terms becomes more crucial as expressions become more complex. Consider this example:

3x² + 5xy - 2x² + 7xy - 4y² + x²

Here, we have several terms with different variables and exponents:

• 3x², -2x², and x² are like terms because they all have x raised to the power of 2.
• 5xy and 7xy are like terms because they both have x and y raised to the power of 1.
• -4y² is a term with y raised to the power of 2; there are no other like terms for this one in this expression.

The Process of Combining Like Terms

The process of combining like terms involves adding or subtracting the coefficients of the like terms while keeping the variable part unchanged.

Let's return to our original expression: 1.3b + 7.8 - 3.2b

1. Identify like terms: We've already identified 1.3b and -3.2b as like terms.

2. Combine the coefficients: Add the coefficients of the like terms: 1.3 + (-3.2) = -1.9

3. Keep the variable part: The variable part remains 'b'.

4. Write the simplified expression: The simplified expression is -1.9b + 7.8

Therefore, 1.3b + 7.8 - 3.2b simplifies to -1.9b + 7.8.

Let's tackle the more complex example: 3x² + 5xy - 2x² + 7xy - 4y² + x²

1. Group like terms: Rearrange the expression to group like terms together: (3x² - 2x² + x²) + (5xy + 7xy) - 4y²

2. Combine coefficients of like terms:

   • For the x² terms: 3 - 2 + 1 = 2
   • For the xy terms: 5 + 7 = 12
   • The -4y² term remains unchanged as it has no like terms.

3. Write the simplified expression: The simplified expression is 2x² + 12xy - 4y²

Common Mistakes to Avoid

• Combining unlike terms: This is the most common mistake. Remember, you can only combine terms with the exact same variables raised to the same powers. For example, you cannot combine 2x and 3y.

• Incorrectly handling signs: Pay close attention to the signs (positive or negative) of the coefficients. Subtracting a negative number is the same as adding a positive number. For example, 5x - (-2x) simplifies to 7x.

• Forgetting to include the variable part: When combining like terms, don't forget to keep the variable part unchanged. The variable part is what makes the terms "like."

• Not simplifying completely: Always check if the simplified expression can be further simplified. For instance, if you end up with 4x + 2x + 6, remember to combine the like terms 4x and 2x to get 6x + 6.

Applications of Combining Like Terms

Combining like terms is essential in various algebraic applications, including:

• Solving equations: Simplifying equations before solving them makes the process easier and less prone to errors.

• Simplifying expressions: Combining like terms makes expressions easier to understand and work with. This is critical in calculus and other advanced mathematical areas.

• Graphing: Simplifying equations using the techniques of combining like terms often makes it much easier to represent the equation graphically.

• Real-world problem-solving: Many real-world problems, especially those involving measurement, finances, and physics, can be modeled using algebraic expressions which benefit greatly from the process of combining like terms.

Practice Problems

To solidify your understanding, try simplifying the following expressions by combining like terms:

1. 4a + 6b - 2a + 3b
2. 5x² - 3x + 2x² + 7x - 8
3. 2m³n + 4mn² - 3m³n + 5mn²
4. -1.5p + 2.7q + 3.2p - 1.8q
5. 7ab² + 3a²b - 2ab² - a²b + 5a²b

Conclusion

Combining like terms is a fundamental algebraic skill that simplifies expressions and makes solving equations more efficient. By mastering this technique and understanding how to identify like terms, correctly handle signs, and avoid common errors, you'll significantly improve your ability to manipulate algebraic expressions and solve a wide range of mathematical problems. Remember the core principle: only combine terms that have the same variables raised to the same powers. Consistent practice will build your confidence and proficiency in this essential algebraic concept.