Combine Like Terms To Create An Equivalent Expression. 0.25k+1.5-k-3.5

Combining Like Terms: A Comprehensive Guide to Simplifying Algebraic Expressions

Combining like terms is a fundamental concept in algebra, essential for simplifying expressions and solving equations. It's a process that involves identifying and then combining terms that share the same variable and exponent. Mastering this skill will significantly improve your ability to manipulate algebraic expressions and solve more complex mathematical problems. This comprehensive guide will walk you through the process, providing examples and tackling various scenarios, including the example expression: 0.25k + 1.5 - k - 3.5.

Understanding Like Terms

Before diving into the mechanics of combining like terms, let's clarify what constitutes "like terms." Like terms are terms in an algebraic expression that have the same variable raised to the same power. The numerical coefficient (the number in front of the variable) can be different, but the variable and its exponent must be identical.

Examples of Like Terms:

• 3x and -5x (same variable 'x', same exponent 1)
• 2y² and 7y² (same variable 'y', same exponent 2)
• -4a³b and 6a³b (same variables 'a' and 'b', same exponents 3 and 1 respectively)

Examples of Unlike Terms:

• 2x and 2y (different variables)
• 4x² and 4x (same variable, different exponents)
• 5ab and 5a²b (same variables, different exponents)

The Process of Combining Like Terms

The process of combining like terms is essentially addition and subtraction. You combine the coefficients of like terms while keeping the variable and exponent unchanged.

Step-by-Step Guide:

1. Identify Like Terms: Carefully examine the expression and identify all terms that have the same variable raised to the same power. It's helpful to underline or circle like terms to make them visually distinct.

2. Group Like Terms: Rearrange the expression, grouping like terms together. This often involves using the commutative property of addition, which states that the order of terms in addition doesn't affect the sum.

3. Combine Coefficients: Add or subtract the coefficients of each group of like terms. Remember to pay attention to the signs (positive or negative) of the coefficients.

4. Write the Simplified Expression: Write the simplified expression, including the combined coefficients and the common variable and exponent for each group of like terms.

Example: Simplifying 0.25k + 1.5 - k - 3.5

Let's apply these steps to the expression 0.25k + 1.5 - k - 3.5.

1. Identify Like Terms: We have two types of like terms:

   • Terms with the variable 'k': 0.25k and -k
   • Constant terms (terms without variables): 1.5 and -3.5

2. Group Like Terms: Rearrange the expression: 0.25k - k + 1.5 - 3.5

3. Combine Coefficients:

   • For the 'k' terms: 0.25k - k = 0.25k - 1k = (0.25 - 1)k = -0.75k
   • For the constant terms: 1.5 - 3.5 = -2

4. Write the Simplified Expression: The simplified expression is -0.75k - 2.

More Complex Examples

Let's explore more intricate examples to solidify your understanding.

Example 1: Simplify 3x² + 5x - 2x² + 7x - 4

1. Identify Like Terms:

   • 3x² and -2x²
   • 5x and 7x
   • -4 (constant term)

2. Group Like Terms: 3x² - 2x² + 5x + 7x - 4

3. Combine Coefficients:

   • 3x² - 2x² = x²
   • 5x + 7x = 12x

4. Simplified Expression: x² + 12x - 4

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

1. Identify Like Terms:

   • 4ab and -3ab
   • 2a and -a
   • 5b (no other like terms)

2. Group Like Terms: 4ab - 3ab + 2a - a + 5b

3. Combine Coefficients:

   • 4ab - 3ab = ab
   • 2a - a = a

4. Simplified Expression: ab + a + 5b

Example 3: Simplify 2x³ + 5x²y - x³ + 3x²y + 4

1. Identify Like Terms:

   • 2x³ and -x³
   • 5x²y and 3x²y
   • 4 (constant term)

2. Group Like Terms: 2x³ - x³ + 5x²y + 3x²y + 4

3. Combine Coefficients:

   • 2x³ - x³ = x³
   • 5x²y + 3x²y = 8x²y

4. Simplified Expression: x³ + 8x²y + 4

Common Mistakes to Avoid

While combining like terms seems straightforward, several common mistakes can lead to incorrect results. Be aware of these pitfalls:

• Ignoring Signs: Pay close attention to the signs (+ or -) in front of each term. A negative sign affects the coefficient of the entire term.

• Incorrectly Identifying Like Terms: Make sure you accurately identify terms with the same variable and exponent.

• Errors in Arithmetic: Double-check your addition and subtraction to prevent simple calculation errors.

• Not simplifying completely: Always ensure you've combined all possible like terms to arrive at the most simplified expression.

Practical Applications

Combining like terms is a fundamental skill with broad applications across various fields that utilize mathematics:

• Solving Equations: Simplifying equations before solving makes them much easier to manage.

• Calculating Areas and Volumes: Many geometric formulas involve combining like terms to determine the final area or volume.

• Data Analysis: Combining like terms is crucial in simplifying and interpreting data.

• Physics and Engineering: Numerous physics and engineering problems rely on simplifying algebraic expressions.

By consistently practicing and understanding the nuances of combining like terms, you'll build a strong foundation in algebra and improve your problem-solving skills in various mathematical contexts. Remember to break down complex expressions into smaller, manageable parts, and always double-check your work to ensure accuracy. Mastering this skill is a crucial step in advancing your mathematical abilities.