Combine Like Terms To Create An Equivalent Expression.-3.6-1.9t+1.2+5.1t

Combining Like Terms: A Comprehensive Guide to Simplifying Algebraic Expressions

Simplifying algebraic expressions is a fundamental skill in algebra. One of the most crucial steps in simplification involves combining like terms. This process streamlines complex expressions, making them easier to understand and manipulate. This comprehensive guide will delve into the intricacies of combining like terms, using the example expression -3.6 - 1.9t + 1.2 + 5.1t to illustrate the concepts. We'll cover the definition of like terms, the steps involved in combining them, and various practical applications.

Understanding Like Terms

Before we dive into combining like terms, it's crucial to understand what constitutes a "like term." 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.

Let's break down the example expression: -3.6 - 1.9t + 1.2 + 5.1t

In this expression, we have four terms:

• -3.6: This is a constant term; it doesn't have any variables.
• -1.9t: This is a term with the variable 't' raised to the power of 1 (remember, t is the same as t¹).
• 1.2: Another constant term.
• 5.1t: This term also contains the variable 't' raised to the power of 1.

Notice that -1.9t and 5.1t are like terms because they both have the same variable (t) raised to the same power (1). Similarly, -3.6 and 1.2 are like terms because they are both constants.

Steps to Combine Like Terms

The process of combining like terms is straightforward:

1. Identify Like Terms: The first step is to identify all the terms in the expression that are alike. This involves carefully examining each term and noting its variable and exponent. In our example, we've already identified the like terms: -1.9t and 5.1t, and -3.6 and 1.2.

2. Group Like Terms: Next, group the like terms together. This makes the process of combining them much easier. We can rewrite our example expression as:

   (-3.6 + 1.2) + (-1.9t + 5.1t)

3. Combine the Coefficients: Now, combine the coefficients of the like terms. Remember that you are only adding or subtracting the coefficients; the variable and its exponent remain unchanged.

   For the constant terms: -3.6 + 1.2 = -2.4 For the 't' terms: -1.9t + 5.1t = 3.2t

4. Write the Simplified Expression: Finally, write the simplified expression by combining the results from step 3. Our simplified expression for -3.6 - 1.9t + 1.2 + 5.1t is:

   -2.4 + 3.2t

This simplified expression is equivalent to the original expression but is much more concise and easier to work with.

Illustrative Examples with Different Scenarios

Let's explore some more examples to solidify our understanding of combining like terms, showcasing different scenarios that you might encounter:

Example 1: More Complex Expressions

Simplify the expression: 5x² + 2xy - 3x² + 7xy - 4y²

1. Identify Like Terms: We have x² terms, xy terms, and y² terms.

2. Group Like Terms: (5x² - 3x²) + (2xy + 7xy) - 4y²

3. Combine Coefficients: 2x² + 9xy - 4y²

4. Simplified Expression: 2x² + 9xy - 4y²

Example 2: Expressions with Negative Coefficients

Simplify: -7a + 3b - 2a - 5b + 10

1. Identify Like Terms: 'a' terms, 'b' terms, and constant terms.

2. Group Like Terms: (-7a - 2a) + (3b - 5b) + 10

3. Combine Coefficients: -9a - 2b + 10

4. Simplified Expression: -9a - 2b + 10

Example 3: Expressions with Fractions

Simplify: (1/2)x + (2/3)y - (1/4)x + (1/6)y

1. Identify Like Terms: 'x' terms and 'y' terms.

2. Group Like Terms: ((1/2)x - (1/4)x) + ((2/3)y + (1/6)y)

3. Combine Coefficients: To add or subtract fractions, find a common denominator. (2/4)x - (1/4)x = (1/4)x (4/6)y + (1/6)y = (5/6)y

4. Simplified Expression: (1/4)x + (5/6)y

The Importance of Combining Like Terms

Combining like terms is not merely a mathematical exercise; it plays a vital role in various algebraic processes:

• Solving Equations: Simplifying equations by combining like terms makes them easier to solve. A simplified equation is easier to understand and manipulate, leading to a more efficient solution.

• Graphing Equations: Combining like terms often leads to a simplified form of an equation, making it easier to identify key features for graphing, such as intercepts and slopes.

• Factorization: Combining like terms can be a preliminary step in factoring expressions, a crucial technique for solving quadratic equations and simplifying complex expressions.

• Problem Solving: In real-world applications of algebra, combining like terms allows for the simplification of complex relationships, providing a clearer understanding of the problem at hand. For example, in physics, combining like terms helps simplify equations representing forces or motion.

Common Mistakes to Avoid

While combining like terms is relatively straightforward, there are some common mistakes to watch out for:

• Adding or Subtracting Unlike Terms: A common mistake is trying to combine unlike terms. Remember, you can only combine terms with the same variable raised to the same power.

• Incorrectly Combining Coefficients: Ensure you are correctly adding or subtracting the coefficients, paying close attention to positive and negative signs.

• Ignoring Exponents: Make sure you are correctly accounting for exponents when identifying like terms. x² and x are not like terms.

• Not Grouping Like Terms: While not strictly an error, failing to group like terms before combining them can lead to confusion and mistakes, especially when working with more complex expressions.

Conclusion

Mastering the skill of combining like terms is essential for success in algebra and beyond. By following the steps outlined in this guide and paying close attention to detail, you can confidently simplify even the most complex algebraic expressions, paving the way for a deeper understanding of mathematical concepts and their practical applications. Remember to always practice, identify and learn from your mistakes, and you'll quickly become proficient in this fundamental algebraic skill. This proficiency will undoubtedly enhance your problem-solving abilities and provide a solid foundation for more advanced mathematical concepts.