Solving 2(-4x + 4) - 4x - 2 = -30 A Step-by-Step Guide

Introduction

In this article, we will delve into the step-by-step process of solving the algebraic equation 2(-4x + 4) - 4x - 2 = -30. This equation involves simplifying expressions, applying the distributive property, combining like terms, and isolating the variable x to find its value. We will break down each step in detail, providing explanations and justifications along the way to ensure a clear understanding of the solution process. This comprehensive guide aims to equip you with the necessary skills to tackle similar algebraic problems with confidence. Understanding how to solve linear equations is a fundamental skill in mathematics, serving as a cornerstone for more advanced topics such as calculus, linear algebra, and differential equations. Therefore, mastering these basic algebraic techniques is crucial for success in higher-level mathematics and various applications in science, engineering, and economics. We will explore the underlying principles of equation solving and demonstrate how to apply these principles systematically to arrive at the correct solution. By the end of this article, you will not only be able to solve the given equation but also gain a deeper appreciation for the logic and elegance of algebraic manipulation. Our approach will emphasize clarity and precision, making the concepts accessible to learners of all levels. Whether you are a student preparing for an exam or simply seeking to refresh your algebraic skills, this article provides a valuable resource for enhancing your mathematical proficiency. We encourage you to follow along with the steps, practice the techniques, and apply your newfound knowledge to solve other equations. With consistent effort and a solid understanding of the principles, you can confidently tackle any algebraic challenge that comes your way.

Step 1: Apply the Distributive Property

The first step in solving the equation 2(-4x + 4) - 4x - 2 = -30 involves applying the distributive property. The distributive property states that a(b + c) = ab + ac. In our case, we need to distribute the 2 across the terms inside the parentheses (-4x + 4). This means we multiply 2 by both -4x and 4.

Applying the distributive property, we get:

2 * (-4x) + 2 * 4 - 4x - 2 = -30

This simplifies to:

-8x + 8 - 4x - 2 = -30

The distributive property is a cornerstone of algebraic manipulation, allowing us to simplify expressions and solve equations that involve parentheses or brackets. Understanding and correctly applying the distributive property is essential for success in algebra and beyond. It's not just about multiplying numbers; it's about understanding the underlying principle of how multiplication interacts with addition and subtraction. This step is crucial because it transforms a more complex expression with parentheses into a simpler form that we can work with more easily. Without the distributive property, we would be stuck trying to deal with the parentheses, which can be a significant obstacle in solving equations. By distributing the 2, we eliminate the parentheses and open the door to combining like terms and further simplifying the equation. The result, -8x + 8 - 4x - 2 = -30, is now in a form where we can clearly see the terms involving x and the constant terms, making the next steps in the solution process more straightforward. Therefore, mastering the distributive property is not just about following a rule; it's about gaining a fundamental tool for simplifying and solving algebraic expressions.

Step 2: Combine Like Terms

After applying the distributive property, we have the equation -8x + 8 - 4x - 2 = -30. The next step is to combine like terms. Like terms are terms that have the same variable raised to the same power. In this equation, we have two terms with x (-8x and -4x) and two constant terms (8 and -2).

Combining the x terms, we add their coefficients: -8x - 4x = -12x.

Combining the constant terms, we add them as well: 8 - 2 = 6.

So, the equation becomes:

-12x + 6 = -30

Combining like terms is a crucial step in simplifying algebraic expressions and equations. It helps to consolidate terms and make the equation easier to solve. By grouping together terms that share the same variable and exponent, we reduce the complexity of the expression and bring clarity to the underlying structure of the equation. In this specific case, combining -8x and -4x into -12x, and combining 8 and -2 into 6, allows us to rewrite the equation in a much more manageable form: -12x + 6 = -30. This simplified form highlights the relationship between the variable x and the constants, making it easier to isolate x and find its value. Without this step, the equation would remain cluttered and less transparent, potentially leading to errors in the subsequent steps. The ability to combine like terms efficiently is a fundamental skill in algebra, enabling us to handle complex expressions with confidence and precision. It's not merely a mechanical process; it reflects a deeper understanding of the structure of algebraic expressions and the principles of arithmetic operations. Therefore, mastering this technique is essential for anyone seeking to excel in algebra and beyond. By practicing combining like terms in various contexts, we develop a keen eye for identifying similar terms and an intuitive sense for simplifying expressions effectively.

Step 3: Isolate the Variable Term

Now that we have the simplified equation -12x + 6 = -30, our next goal is to isolate the variable term (-12x). To do this, we need to eliminate the constant term (+6) from the left side of the equation. We can achieve this by subtracting 6 from both sides of the equation. Remember, to maintain the balance of the equation, we must perform the same operation on both sides.

Subtracting 6 from both sides, we get:

-12x + 6 - 6 = -30 - 6

This simplifies to:

-12x = -36

Isolating the variable term is a critical step in solving any algebraic equation. It's the process of getting the term containing the variable (in this case, -12x) by itself on one side of the equation. This allows us to focus on solving for the variable in the next step. The principle behind this step is the idea of maintaining equality. Whatever operation we perform on one side of the equation, we must also perform on the other side to ensure that the equation remains balanced. In this specific instance, we subtracted 6 from both sides to cancel out the +6 on the left side, effectively isolating the -12x term. This step transforms the equation from -12x + 6 = -30 to -12x = -36, which is a much simpler form to work with. The isolation of the variable term sets the stage for the final step, where we will divide both sides by the coefficient of x to solve for the value of x. Mastering the technique of isolating the variable term is fundamental to solving algebraic equations, and it's a skill that will be used repeatedly in more advanced mathematical contexts. It requires a clear understanding of the properties of equality and the ability to manipulate equations strategically to achieve the desired outcome.

Step 4: Solve for x

We have now arrived at the equation -12x = -36. To solve for x, we need to isolate x completely. This means we need to get rid of the coefficient -12 that is multiplying x. We can do this by dividing both sides of the equation by -12.

Dividing both sides by -12, we get:

-12x / -12 = -36 / -12

This simplifies to:

x = 3

Therefore, the solution to the equation is x = 3. Solving for the variable is the ultimate goal in solving any algebraic equation. It's the process of finding the value of the variable that makes the equation true. In this case, we had the equation -12x = -36, and we needed to undo the multiplication by -12 to isolate x. The principle we used is the inverse operation: division. Dividing both sides of the equation by -12 cancels out the multiplication by -12 on the left side, leaving us with x by itself. On the right side, -36 divided by -12 equals 3. So, we arrive at the solution x = 3. This means that if we substitute 3 for x in the original equation, the equation will hold true. The ability to solve for a variable is a fundamental skill in algebra, and it's the foundation for solving more complex equations and problems in mathematics and various scientific disciplines. It requires a clear understanding of the inverse operations and the ability to apply them strategically to isolate the variable. Mastering this technique empowers us to find solutions to a wide range of mathematical problems and to make predictions and decisions based on mathematical models.

Verification

To ensure the accuracy of our solution, it's always a good practice to verify it. We can do this by substituting the value we found for x (which is 3) back into the original equation and checking if the equation holds true.

Original equation: 2(-4x + 4) - 4x - 2 = -30

Substitute x = 3:

2(-4(3) + 4) - 4(3) - 2 = -30

Simplify:

2(-12 + 4) - 12 - 2 = -30

2(-8) - 12 - 2 = -30

-16 - 12 - 2 = -30

-30 = -30

Since the equation holds true, our solution x = 3 is correct.

Verification is an essential step in the problem-solving process, particularly in mathematics. It's the process of checking whether the solution we have obtained actually satisfies the original equation or problem conditions. In this case, we substituted the value x = 3 back into the original equation and performed the arithmetic operations to see if both sides of the equation were equal. The verification process serves several important purposes. First, it helps us to identify any errors we might have made in the solution process. By substituting the solution back into the original equation, we can catch mistakes in arithmetic, algebraic manipulation, or logical reasoning. Second, verification gives us confidence in our solution. When we confirm that our solution satisfies the original equation, we can be sure that we have found the correct answer. Third, verification reinforces our understanding of the problem and the solution process. It helps us to see how the different parts of the equation or problem fit together and how the solution addresses the problem's requirements. In this specific case, the verification process showed that when we substitute x = 3 into the original equation, both sides are equal to -30. This confirms that our solution is correct and that we have successfully solved the equation. The habit of verifying solutions is a valuable one to cultivate in mathematics and other fields, as it promotes accuracy, confidence, and a deeper understanding of the problem-solving process.

Conclusion

In this article, we have provided a detailed walkthrough of solving the equation 2(-4x + 4) - 4x - 2 = -30. We have demonstrated how to apply the distributive property, combine like terms, isolate the variable term, and solve for x. We also emphasized the importance of verifying the solution to ensure its accuracy. By following these steps carefully, you can confidently solve similar algebraic equations.

Mastering the process of solving algebraic equations is a fundamental skill in mathematics, and it is essential for success in higher-level courses and various applications in science, engineering, and economics. The ability to manipulate equations, isolate variables, and find solutions is a powerful tool that can be applied to a wide range of problems. The steps we have outlined in this article—applying the distributive property, combining like terms, isolating the variable term, and solving for the variable—form the core of algebraic equation solving. By practicing these steps and understanding the underlying principles, you can develop a strong foundation in algebra. Furthermore, the emphasis on verification underscores the importance of accuracy and attention to detail in mathematical problem-solving. Checking our solutions not only helps us to identify errors but also deepens our understanding of the problem and the solution process. In conclusion, the techniques and principles discussed in this article provide a solid foundation for solving algebraic equations and for approaching more complex mathematical challenges with confidence. By continuing to practice and apply these skills, you can enhance your mathematical proficiency and unlock new opportunities for learning and problem-solving.