Simplifying Algebraic Expressions 5(-2.3b - 4) - (7b + 9)

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#Title: Simplifying Algebraic Expressions A Step-by-Step Solution

In the realm of mathematics, simplifying algebraic expressions is a fundamental skill. This article will delve into the process of simplifying the expression 5(-2.3b - 4) - (7b + 9), providing a step-by-step solution and exploring the underlying mathematical principles. Mastering this skill is crucial for success in algebra and beyond, as it forms the basis for solving equations, inequalities, and more complex mathematical problems. Let's embark on this journey of algebraic simplification together!

Understanding the Expression: 5(-2.3b - 4) - (7b + 9)

Before we jump into the simplification process, let's take a closer look at the expression itself: 5(-2.3b - 4) - (7b + 9). This expression involves several key components:

  • Variables: The variable in this expression is 'b', representing an unknown quantity.
  • Coefficients: The numbers multiplied by the variable are called coefficients. In this case, we have -2.3 as a coefficient for 'b' within the parentheses and 7 as a coefficient for 'b' in the second set of parentheses.
  • Constants: The numbers that stand alone are called constants. We have -4 within the first set of parentheses and 9 in the second set.
  • Operations: The expression involves multiplication (5 multiplied by the expression in parentheses), subtraction (between the two sets of parentheses), and addition (within the parentheses).

To simplify this expression effectively, we need to understand the order of operations and how to apply the distributive property. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which we perform mathematical operations. The distributive property allows us to multiply a number by a sum or difference within parentheses.

Step-by-Step Simplification

Now, let's break down the simplification process into manageable steps:

1. Apply the Distributive Property

The first step is to distribute the 5 across the terms inside the first set of parentheses: 5(-2.3b - 4). This means we multiply 5 by both -2.3b and -4:

  • 5 * -2.3b = -11.5b
  • 5 * -4 = -20

So, the expression becomes: -11.5b - 20 - (7b + 9).

The next step is to distribute the negative sign (which is the same as multiplying by -1) across the terms inside the second set of parentheses: -(7b + 9):

  • -1 * 7b = -7b
  • -1 * 9 = -9

Now, the expression looks like this: -11.5b - 20 - 7b - 9.

2. Combine Like Terms

Like terms are terms that have the same variable raised to the same power. In our expression, the like terms are the 'b' terms (-11.5b and -7b) and the constant terms (-20 and -9). Let's combine them:

  • Combine the 'b' terms: -11.5b - 7b = -18.5b
  • Combine the constant terms: -20 - 9 = -29

After combining like terms, the expression simplifies to: -18.5b - 29.

The Equivalent Expression

Therefore, the simplified expression, which is equivalent to the original expression 5(-2.3b - 4) - (7b + 9), is -18.5b - 29. This corresponds to option A in the original question.

Why is Simplifying Expressions Important?

Simplifying algebraic expressions is not just a mathematical exercise; it's a crucial skill with numerous applications in various fields. Here are some reasons why it's important:

  • Solving Equations: Simplified expressions make it easier to solve equations. By reducing an expression to its simplest form, we can isolate the variable and find its value more efficiently.
  • Problem-Solving: Many real-world problems can be modeled using algebraic expressions. Simplifying these expressions helps us understand the relationships between variables and find solutions.
  • Further Mathematical Studies: Simplifying expressions is a fundamental skill for more advanced mathematical topics such as calculus, linear algebra, and differential equations.
  • Computer Science: In computer programming, simplifying expressions is essential for optimizing code and improving performance.

Common Mistakes to Avoid

While simplifying expressions, it's easy to make mistakes. Here are some common pitfalls to watch out for:

  • Incorrectly Applying the Distributive Property: Make sure to multiply the term outside the parentheses by every term inside the parentheses.
  • Forgetting the Negative Sign: When distributing a negative sign, remember to change the sign of every term inside the parentheses.
  • Combining Unlike Terms: Only combine terms that have the same variable raised to the same power.
  • Order of Operations Errors: Always follow the order of operations (PEMDAS) to ensure correct simplification.

By being aware of these common mistakes, you can improve your accuracy and avoid errors in your calculations.

Practice Problems

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

  1. Simplify: 3(2x + 5) - (x - 2)
  2. Simplify: -2(4y - 1) + 6y
  3. Simplify: 7(a + 3) - 4(2a - 1)
  4. Simplify: -5(3z - 2) + 8z - 1

Working through these problems will help you practice the steps involved in simplifying expressions and build your confidence in your abilities.

Conclusion

In conclusion, simplifying algebraic expressions is a fundamental skill in mathematics with far-reaching applications. By understanding the distributive property, combining like terms, and avoiding common mistakes, you can master this skill and unlock the door to more advanced mathematical concepts. The expression 5(-2.3b - 4) - (7b + 9) simplifies to -18.5b - 29, demonstrating the power of algebraic manipulation. Keep practicing, and you'll become a pro at simplifying expressions!