Simplifying Algebraic Expressions Combining Like Terms In 20(-1.5 R+0.75)

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In the realm of mathematics, simplifying algebraic expressions is a fundamental skill. It involves manipulating expressions to make them easier to understand and work with. One key technique in this process is combining like terms. In this comprehensive exploration, we will delve into the intricacies of combining like terms, using the expression 20(-1.5 r + 0.75) as our guiding example. Our journey will cover the basic principles, step-by-step simplification, and the underlying logic that makes this technique so powerful. This article will serve as an indispensable resource for students, educators, and anyone looking to sharpen their algebra skills.

Understanding Like Terms

To effectively combine like terms, we must first define what they are. Like terms are terms that have the same variable(s) raised to the same power(s). Only the coefficients (the numbers in front of the variables) can differ. For instance, 3x and -5x are like terms because they both contain the variable x raised to the power of 1. Similarly, 2y^2 and 7y^2 are like terms because they both have y raised to the power of 2. However, 4x and 4x^2 are not like terms because the variable x is raised to different powers.

Constant terms, which are numbers without any variables (e.g., 5, -2, 0.75), are also considered like terms. This is because they can be thought of as having a variable raised to the power of 0 (since any number to the power of 0 is 1). Therefore, combining like terms often involves adding or subtracting the coefficients of terms that share the same variable and exponent, or simply adding or subtracting constant terms.

Recognizing like terms is the bedrock of simplifying algebraic expressions. It allows us to consolidate multiple terms into a single, more manageable term. This simplification not only makes expressions easier to read but also facilitates further algebraic manipulations, such as solving equations or evaluating expressions for specific values of the variables.

The Distributive Property: A Key Tool

Before we can combine like terms in the given expression, 20(-1.5 r + 0.75), we need to address the parentheses. This is where the distributive property comes into play. The distributive property states that for any numbers a, b, and c:

a(b + c) = ab + ac

In simpler terms, this means we can multiply the term outside the parentheses by each term inside the parentheses. This property is crucial for expanding expressions and removing parentheses, which is often a necessary step before combining like terms.

Applying the distributive property to our expression, we multiply 20 by both -1.5 r and 0.75:

20 * (-1.5 r) + 20 * 0.75

Now we perform the multiplications:

-30r + 15

With the parentheses removed, the expression is now in a form where we can easily identify and combine like terms. This step highlights the importance of the distributive property in simplifying algebraic expressions. It allows us to transform expressions into a more accessible format for further manipulation.

Step-by-Step Simplification of 20(-1.5 r + 0.75)

Now that we've laid the groundwork, let's walk through the simplification of the expression 20(-1.5 r + 0.75) step-by-step:

  1. Apply the Distributive Property:

    • As we discussed earlier, the first step is to distribute the 20 across the terms inside the parentheses:
    • 20 * (-1.5 r) + 20 * 0.75

  2. Perform the Multiplications:

    • Next, we carry out the multiplication operations:
    • -30r + 15

  3. Identify Like Terms:

    • In this simplified expression, we have two terms: -30r and 15.
    • -30r is a term with the variable r, and 15 is a constant term.

  4. Combine Like Terms (if any):

    • In this case, -30r and 15 are not like terms because one has the variable r and the other is a constant. Therefore, they cannot be combined further.

  5. Final Simplified Expression:

    • The final simplified expression is: -30r + 15

This step-by-step approach illustrates how we systematically transform the original expression into its simplest form. Each step is grounded in fundamental algebraic principles, ensuring accuracy and clarity in the simplification process.

Practical Examples and Further Applications

To solidify our understanding, let's explore additional examples of combining like terms:

  • Example 1: Simplify 3x + 2y - 5x + y

    • Identify like terms: 3x and -5x are like terms; 2y and y are like terms.
    • Combine like terms: (3x - 5x) + (2y + y) = -2x + 3y

  • Example 2: Simplify 4a^2 - 2a + 7 - a^2 + 3a - 2

    • Identify like terms: 4a^2 and -a^2 are like terms; -2a and 3a are like terms; 7 and -2 are like terms.
    • Combine like terms: (4a^2 - a^2) + (-2a + 3a) + (7 - 2) = 3a^2 + a + 5

These examples demonstrate the versatility of combining like terms in various algebraic expressions. This technique is not just an isolated skill; it's a building block for more advanced algebraic concepts, such as solving equations, factoring polynomials, and simplifying rational expressions.

Furthermore, the ability to simplify expressions has practical applications in various fields. In physics, it can help simplify equations that describe the motion of objects. In engineering, it can be used to optimize designs and calculations. In computer science, it plays a role in algorithm optimization. Mastering this skill opens doors to a deeper understanding of mathematical concepts and their real-world applications.

Common Mistakes to Avoid

While combining like terms is a straightforward process, there are common pitfalls that students often encounter. Being aware of these mistakes can help prevent errors and ensure accurate simplification.

  1. Combining Unlike Terms: This is perhaps the most frequent mistake. It involves adding or subtracting terms that do not have the same variable and exponent. For example, incorrectly combining 2x and 3x^2. Remember, only terms with the same variable raised to the same power can be combined.

  2. Forgetting the Distributive Property: When an expression contains parentheses, it's crucial to apply the distributive property correctly. Failing to do so can lead to significant errors. Make sure to multiply the term outside the parentheses by each term inside.

  3. Incorrectly Handling Coefficients: When combining like terms, pay close attention to the coefficients. Ensure that you are correctly adding or subtracting them. A simple arithmetic error can change the entire result.

  4. Ignoring Signs: Terms often have negative signs in front of them. It's essential to carry these signs along when combining like terms. For example, in the expression 5x - 3x, the - sign in front of 3x is crucial.

By being mindful of these common mistakes, you can improve your accuracy and confidence in simplifying algebraic expressions.

Conclusion: The Power of Simplification

In conclusion, combining like terms is a fundamental technique in algebra that simplifies expressions and makes them easier to work with. By understanding the concept of like terms, applying the distributive property, and following a systematic approach, you can confidently simplify a wide range of algebraic expressions. The simplification of the expression 20(-1.5 r + 0.75) to -30r + 15 exemplifies this process.

This skill is not just about manipulating symbols; it's about developing a deeper understanding of mathematical structures and relationships. It lays the foundation for more advanced topics in algebra and beyond. Whether you're a student learning algebra for the first time or someone looking to refresh your skills, mastering the art of combining like terms is an investment in your mathematical journey. The ability to simplify expressions efficiently and accurately will serve you well in various mathematical and real-world contexts.