Combine Like Terms To Create An Equivalent Expression.-5.8c+4.2-3.1+1.4c

Combining Like Terms: A Comprehensive Guide to Simplifying Algebraic Expressions

Simplifying algebraic expressions is a fundamental skill in mathematics. A key step in this process is combining like terms. This article will delve deep into the concept of combining like terms, providing a clear understanding of what like terms are, how to identify them, and the step-by-step process involved in simplifying expressions using this technique. We'll explore various examples, including the specific expression -5.8c + 4.2 - 3.1 + 1.4c, and address common pitfalls to avoid. By the end, you'll be confident in your ability to simplify even the most complex algebraic expressions.

Understanding Like Terms

Before diving into the mechanics of combining like terms, let's first establish a clear understanding of what constitutes a "like term." Like terms are terms in an algebraic expression that have the same variable raised to the same power. Consider these examples:

• Like Terms: 3x and 5x (both have the variable 'x' raised to the power of 1)
• Like Terms: -2y² and 7y² (both have the variable 'y' raised to the power of 2)
• Unlike Terms: 4x and 4y (different variables)
• Unlike Terms: 2x and 2x² (same variable, but different powers)
• Like Terms: 6 and -2 (both are constants, meaning they don't have any variables)

The key takeaway here is that both the variable and its exponent must be identical for terms to be considered like terms. Constants are always considered like terms, as they are essentially terms with a variable raised to the power of zero (x⁰ = 1).

The Process of Combining Like Terms

The process of combining like terms is based on the distributive property of multiplication over addition (and subtraction). This property states that a(b + c) = ab + ac. When combining like terms, we are essentially factoring out the common variable and its exponent.

Let's break down the steps involved:

1. Identify Like Terms: Carefully examine the expression and identify all the terms that have the same variable raised to the same power. Group these like terms together.

2. Add or Subtract Coefficients: Once you have grouped the like terms, add or subtract their coefficients (the numbers in front of the variables). Remember that the sign in front of the term belongs to the coefficient.

3. Simplify the Expression: Combine the simplified coefficients with the common variable and its exponent. This resulting expression will be the simplified equivalent expression.

Solving the Example: -5.8c + 4.2 - 3.1 + 1.4c

Now, let's apply this process to the example expression provided: -5.8c + 4.2 - 3.1 + 1.4c

1. Identify Like Terms:

• Like terms with variable 'c': -5.8c and 1.4c
• Like terms (constants): 4.2 and -3.1

2. Add or Subtract Coefficients:

• For the 'c' terms: -5.8c + 1.4c = (-5.8 + 1.4)c = -4.4c
• For the constants: 4.2 - 3.1 = 1.1

3. Simplify the Expression:

Combining the simplified coefficients with their respective variables, we obtain: -4.4c + 1.1

Therefore, the simplified equivalent expression for -5.8c + 4.2 - 3.1 + 1.4c is -4.4c + 1.1.

More Complex Examples

Let's explore some more challenging examples to further solidify your understanding:

Example 1: 3x² + 5x - 2x² + 7x + 9

1. Identify Like Terms:

   • Like terms with x²: 3x² and -2x²
   • Like terms with x: 5x and 7x
   • Constant term: 9

2. Add or Subtract Coefficients:

   • For x² terms: 3x² - 2x² = (3 - 2)x² = x²
   • For x terms: 5x + 7x = (5 + 7)x = 12x

3. Simplify the Expression: The simplified expression is: x² + 12x + 9

Example 2: 4ab - 2a + 6ab + 5a - 3b

1. Identify Like Terms:

   • Like terms with ab: 4ab and 6ab
   • Like terms with a: -2a and 5a
   • Like term with b: -3b

2. Add or Subtract Coefficients:

   • For ab terms: 4ab + 6ab = (4 + 6)ab = 10ab
   • For a terms: -2a + 5a = (-2 + 5)a = 3a

3. Simplify the Expression: The simplified expression is: 10ab + 3a - 3b

Common Mistakes to Avoid

While combining like terms might seem straightforward, several common mistakes can lead to incorrect results. Here are some points to watch out for:

• Incorrectly Identifying Like Terms: Carefully review the variables and their exponents to ensure you're grouping only the like terms. A common mistake is to treat unlike terms as if they were alike.

• Errors in Arithmetic: Double-check your addition and subtraction of coefficients to avoid calculation errors. A simple mistake here can significantly affect the final answer.

• Forgetting the Signs: Pay close attention to the signs (+ or -) in front of each term. These signs are crucial in determining the correct operation (addition or subtraction) when combining coefficients.

• Ignoring the Exponents: Ensure that you're only combining terms with the same variable raised to the exact same power.

Advanced Applications and Extensions

Combining like terms is a fundamental building block for many algebraic manipulations. It forms the basis of solving equations, simplifying complex expressions, and working with polynomials. Mastering this skill is essential for success in higher-level mathematics. As you progress, you'll encounter more intricate expressions involving multiple variables and higher powers. The fundamental principles remain the same, but the complexity increases, demanding careful attention to detail and a thorough understanding of the underlying concepts.

Conclusion

Combining like terms is a powerful tool for simplifying algebraic expressions, making them easier to understand and work with. By following the steps outlined in this comprehensive guide, you'll develop the skills necessary to confidently tackle a wide range of algebraic problems. Remember to always double-check your work, paying close attention to identifying like terms accurately and performing arithmetic calculations correctly. With practice, this seemingly simple skill will become second nature, allowing you to tackle more complex algebraic challenges with ease and confidence. Continue practicing with diverse examples, and you'll master the art of simplifying algebraic expressions through combining like terms.