Simplifying Algebraic Expressions Step-by-Step Solution

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Introduction:

In the realm of mathematics, algebraic expressions serve as fundamental building blocks for more complex equations and formulas. The ability to manipulate and simplify these expressions is a crucial skill for students and professionals alike. This comprehensive guide will delve into the process of simplifying the algebraic expression 3(5x+7)-6(4-2x), providing a step-by-step solution and exploring the underlying mathematical principles.

Understanding the Expression: 3(5x+7)-6(4-2x)

The given algebraic expression is 3(5x+7)-6(4-2x). This expression involves the distributive property, which is a key concept in simplifying algebraic expressions. The distributive property states that for any numbers a, b, and c, a(b+c) = ab + ac. In other words, we multiply the term outside the parentheses by each term inside the parentheses.

Before we dive into the solution, let's break down the expression into smaller parts:

  • 3(5x+7): This part of the expression involves multiplying the term 3 by the binomial (5x+7).
  • -6(4-2x): This part involves multiplying the term -6 by the binomial (4-2x). Note the negative sign in front of the 6, which is crucial for correct distribution.

Step-by-Step Solution:

1. Apply the Distributive Property

Our first step is to apply the distributive property to both parts of the expression:

  • 3(5x+7) = 3 * 5x + 3 * 7 = 15x + 21
  • -6(4-2x) = -6 * 4 + (-6) * (-2x) = -24 + 12x

Remember that multiplying two negative numbers results in a positive number. This is why (-6) * (-2x) becomes +12x.

2. Combine the Results

Now that we've distributed the terms, we can combine the results:

15x + 21 - 24 + 12x

3. Group Like Terms

Next, we group the like terms together. Like terms are terms that have the same variable raised to the same power. In this case, we have two terms with 'x' (15x and 12x) and two constant terms (21 and -24):

(15x + 12x) + (21 - 24)

4. Combine Like Terms

Now we combine the like terms by adding or subtracting their coefficients:

  • 15x + 12x = 27x
  • 21 - 24 = -3

5. Write the Simplified Expression

Finally, we write the simplified expression by combining the results:

27x - 3

Therefore, the algebraic expression equivalent to 3(5x+7)-6(4-2x) is 27x - 3.

Analyzing the Answer Choices

Now, let's examine the answer choices provided:

A. 27x - 12 B. 12x + 27 C. 27x - 3 D. 12x + 3

Comparing our simplified expression (27x - 3) with the answer choices, we can clearly see that option C (27x - 3) is the correct answer.

Common Mistakes and How to Avoid Them

Simplifying algebraic expressions can be tricky, and there are several common mistakes that students often make. Here are a few to watch out for:

  • Incorrectly distributing the negative sign: A common error is forgetting to distribute the negative sign when multiplying a negative term by a binomial. For example, in the expression -6(4-2x), it's crucial to distribute the -6 to both the 4 and the -2x. Failing to do so can lead to an incorrect result.
  • Combining unlike terms: Another common mistake is combining terms that are not like terms. Remember, like terms have the same variable raised to the same power. For example, 2x and 2x² are not like terms and cannot be combined.
  • Arithmetic errors: Simple arithmetic errors, such as adding or subtracting numbers incorrectly, can also lead to incorrect answers. It's always a good idea to double-check your calculations to avoid these errors.

To avoid these mistakes, it's helpful to follow a systematic approach, such as the step-by-step method outlined above. Additionally, practice is key to mastering algebraic simplification. The more you practice, the more comfortable you'll become with the process and the less likely you are to make mistakes.

Why is Simplifying Algebraic Expressions Important?

Simplifying algebraic expressions is a fundamental skill in mathematics with applications in various fields, including:

  • Solving equations: Simplifying expressions is often a necessary step in solving algebraic equations. By simplifying an equation, we can isolate the variable and find its value.
  • Calculus: In calculus, simplifying expressions is essential for finding derivatives and integrals.
  • Physics and Engineering: Many physical and engineering problems involve algebraic equations and formulas. Simplifying these expressions is crucial for solving these problems.
  • Computer Science: Algebraic expressions are used extensively in computer programming and algorithm design.

Mastering the art of simplifying algebraic expressions opens doors to a deeper understanding of mathematical concepts and their applications in various real-world scenarios. It empowers you to tackle complex problems with confidence and precision.

Practice Problems

To solidify your understanding of simplifying algebraic expressions, try solving these practice problems:

  1. Simplify: 2(3x - 5) + 4(x + 2)
  2. Simplify: -5(2y + 1) - 3(y - 4)
  3. Simplify: 4(a - 3) + 2(5a + 1)
  4. Simplify: -2(4b - 2) - (b + 3)
  5. Simplify: 3(2c + 7) - 5(c - 1)

By working through these problems, you'll gain valuable experience and further develop your skills in simplifying algebraic expressions.

Conclusion

In conclusion, simplifying the algebraic expression 3(5x+7)-6(4-2x) involves applying the distributive property, combining like terms, and carefully managing negative signs. By following a step-by-step approach and avoiding common mistakes, you can confidently simplify a wide range of algebraic expressions. Remember, practice is key to mastering this skill, and the ability to simplify expressions is fundamental to success in mathematics and related fields.

The correct answer is C. 27x - 3. This detailed explanation provides a thorough understanding of the simplification process, highlighting the importance of each step and emphasizing the underlying mathematical principles.