Simonne's Guide To Simplifying 12 - 3(-2x + 4) A Step-by-Step Breakdown

JU07/10/2025 08, 2025 by THE IDEN 72 views

Simplifying Algebraic Expressions A Step-by-Step Guide with Example

Introduction

In mathematics, simplifying expressions is a fundamental skill. It allows us to rewrite complex expressions in a more manageable and understandable form. This process often involves applying the distributive property, combining like terms, and following the order of operations. In this article, we will delve into the step-by-step simplification of the expression 12 - 3(-2x + 4), as demonstrated by Simonne. We will break down each step, explain the underlying principles, and highlight the key concepts involved. This comprehensive guide aims to provide a clear understanding of the simplification process, making it accessible to learners of all levels. Whether you are a student learning algebra or someone looking to refresh your mathematical skills, this article will equip you with the knowledge and confidence to tackle similar problems.

Problem Statement

Simonne simplified the algebraic expression 12 - 3(-2x + 4) using a series of steps. Our goal is to understand and analyze each step to grasp the complete simplification process. The initial expression presents a combination of constants, variables, and parentheses, which requires a systematic approach to simplify correctly. By examining Simonne's method, we can gain insights into the strategies used for simplifying such expressions. This problem serves as a practical example for learning how to apply the distributive property and combine like terms, which are essential skills in algebra. Throughout this article, we will explore each step in detail, ensuring a thorough understanding of the mathematical principles involved.

Step-by-Step Simplification

Step 1: Applying the Distributive Property

The first step in simplifying the expression 12 - 3(-2x + 4) involves applying the distributive property. The distributive property states that for any numbers a, b, and c, a(b + c) = ab + ac. In our case, we need to distribute the -3 across the terms inside the parentheses (-2x and +4). This means we multiply -3 by both -2x and +4. Simonne correctly rewrites the expression as 12 + (-3)(-2x) + (-3)(4). This step is crucial because it eliminates the parentheses, allowing us to combine like terms in subsequent steps. By understanding and applying the distributive property, we transform the expression into a form that is easier to simplify. This step demonstrates a fundamental principle in algebra, and mastering it is essential for simplifying more complex expressions. The proper application of the distributive property ensures that we maintain the equality of the expression while making it more manageable.

Step 2: Performing the Multiplication

In the second step, we perform the multiplication operations resulting from the distributive property applied in Step 1. Simonne's expression 12 + (-3)(-2x) + (-3)(4) now requires us to multiply -3 by -2x and -3 by 4. Recall that the product of two negative numbers is a positive number. Therefore, (-3)(-2x) equals 6x. Next, we multiply -3 by 4, which gives us -12. Thus, the expression becomes 12 + 6x + (-12). This step is crucial because it simplifies the terms within the expression, bringing us closer to combining like terms. By accurately performing the multiplication, we reduce the complexity of the expression and prepare it for further simplification. This step highlights the importance of understanding the rules of multiplication, particularly when dealing with negative numbers and variables. The correct execution of this step is vital for arriving at the correct final simplified expression.

Step 3: Rearranging Terms

Step 3 involves rearranging the terms in the expression 12 + 6x + (-12). The purpose of rearranging terms is to group like terms together, making it easier to combine them in the next step. Like terms are terms that have the same variable raised to the same power, or constants. In this case, 12 and -12 are like terms. Simonne rearranges the expression as 12 + (-12) + 6x. By placing the constants next to each other, we can easily see that they will cancel out. This step is an application of the commutative property of addition, which states that the order in which numbers are added does not change the sum. Rearranging terms is a useful technique in simplifying expressions, as it allows us to visually organize the expression and identify terms that can be combined. This step prepares the expression for the final simplification by grouping the constants together.

Step 4: Combining Like Terms

In Step 4, we combine the like terms in the expression 12 + (-12) + 6x. As identified in the previous step, 12 and -12 are like terms. When we add 12 and -12, the result is 0. Therefore, the expression simplifies to 0 + 6x. This step is a straightforward application of addition and subtraction, but it is crucial in reducing the expression to its simplest form. Combining like terms is a fundamental skill in algebra, and it involves adding or subtracting coefficients of terms with the same variable and exponent. In this case, the constants 12 and -12 cancel each other out, leaving only the term with the variable x. This step demonstrates the efficiency of combining like terms in simplifying expressions and highlights the importance of accurate arithmetic in algebraic manipulations. The result of this step brings us one step closer to the final simplified expression.

Step 5: Final Simplified Expression

The final step in Simonne's simplification process is to arrive at the simplest form of the expression. From Step 4, we have 0 + 6x. Adding 0 to any quantity does not change the quantity. Therefore, 0 + 6x simplifies to 6x. This is the final simplified expression. This step demonstrates the additive identity property, which states that adding 0 to any number does not change the number. The simplified expression 6x is much easier to understand and work with than the original expression 12 - 3(-2x + 4). This final step underscores the goal of simplification: to rewrite an expression in its most concise and understandable form. By following the steps of applying the distributive property, performing multiplication, rearranging terms, and combining like terms, we have successfully simplified the given algebraic expression.

Conclusion

In conclusion, Simonne's step-by-step simplification of the expression 12 - 3(-2x + 4) provides a clear illustration of the principles of algebraic manipulation. By applying the distributive property, performing multiplication, rearranging terms, and combining like terms, we have successfully simplified the expression to 6x. Each step is crucial and builds upon the previous one, demonstrating the importance of a systematic approach in mathematics. This example not only reinforces the fundamental concepts of algebra but also highlights the significance of accuracy and attention to detail in mathematical problem-solving. Understanding these steps equips learners with the necessary skills to tackle similar simplification problems with confidence. The process demonstrated here serves as a valuable tool for anyone looking to enhance their algebraic skills and deepen their understanding of mathematical principles. Through careful analysis and step-by-step execution, we can transform complex expressions into simpler, more manageable forms.