Solving 10 = 2 - 4(ax - 3) For X A Step By Step Guide

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In the realm of mathematics, solving equations is a fundamental skill. This article provides a comprehensive, step-by-step guide on how to solve the equation 10 = 2 - 4(ax - 3) for x. We will delve into the algebraic manipulations required to isolate x and arrive at the correct solution. This detailed explanation will not only help you understand the solution but also enhance your problem-solving skills in algebra. The answer options provided are A. -5/a, B. 1/a, C. 5/a, and D. -1/a. Let's embark on this mathematical journey together and unravel the value of x.

1. Understanding the Equation and the Goal

Before we dive into the solution, let's first understand the equation we are dealing with: 10 = 2 - 4(ax - 3). This is a linear equation in terms of x, where 'a' is a constant. Our ultimate goal is to isolate x on one side of the equation to determine its value. This involves performing algebraic operations on both sides of the equation while maintaining the equality.

To effectively solve this equation, we must follow the order of operations and apply the distributive property correctly. It is also crucial to combine like terms and simplify the equation at each step. By meticulously performing these operations, we will gradually move towards isolating x and finding its value in terms of 'a'. The concepts of inverse operations and maintaining balance on both sides of the equation are also paramount in this process. Solving equations is a cornerstone of algebra, and mastering these techniques will prove invaluable in more advanced mathematical studies.

2. Step-by-Step Solution

Now, let's walk through the step-by-step solution of the equation:

Step 1: Distribute the -4

Our initial equation is 10 = 2 - 4(ax - 3). The first step is to distribute the -4 across the terms inside the parentheses. This gives us:

10 = 2 - 4ax + 12

Step 2: Combine Like Terms

Next, we combine the constant terms on the right side of the equation, which are 2 and 12:

10 = 14 - 4ax

Step 3: Isolate the Term with x

To isolate the term containing x (-4ax), we subtract 14 from both sides of the equation:

10 - 14 = 14 - 4ax - 14

This simplifies to:

-4 = -4ax

Step 4: Solve for x

Finally, to solve for x, we divide both sides of the equation by -4a:

x = -4 / (-4a)

This simplifies to:

x = 1/a

Therefore, the solution to the equation 10 = 2 - 4(ax - 3) is x = 1/a. This step-by-step approach illustrates the methodical process of solving algebraic equations. By understanding each step and the underlying principles, you can confidently tackle similar problems.

3. Verifying the Solution

To ensure our solution is correct, it is always a good practice to verify the solution. We do this by substituting the value we found for x (which is 1/a) back into the original equation:

Original equation: 10 = 2 - 4(ax - 3)

Substitute x = 1/a: 10 = 2 - 4(a(1/a) - 3)

Now, simplify the equation:

10 = 2 - 4(1 - 3)

10 = 2 - 4(-2)

10 = 2 + 8

10 = 10

Since the equation holds true after substituting x = 1/a, our solution is correct. This verification process underscores the importance of checking your work to avoid errors and gain confidence in your answer. It is a crucial step in mathematical problem-solving.

4. Analyzing the Answer Choices

Now, let's analyze the answer choices provided in the question. The options were:

A. -5/a B. 1/a C. 5/a D. -1/a

Based on our step-by-step solution, we found that x = 1/a. Therefore, the correct answer choice is B. 1/a. Analyzing the answer choices after solving the equation helps to ensure that you select the correct option and reinforces your understanding of the solution.

Understanding why the other options are incorrect is also beneficial. Options A, C, and D do not match the value of x we derived through our algebraic manipulations. This highlights the importance of careful calculation and verification in solving equations.

5. Common Mistakes and How to Avoid Them

When solving equations like this, there are common mistakes that students often make. Identifying these pitfalls and learning how to avoid them can significantly improve your accuracy. One common mistake is incorrectly distributing the -4 in the first step. It is essential to remember to multiply -4 by both terms inside the parentheses, including the negative sign. Another frequent error is failing to combine like terms properly, which can lead to an incorrect simplification of the equation.

Additionally, errors can occur when isolating the variable x. For instance, students might add instead of subtract, or divide by the wrong coefficient. To avoid these mistakes, it is crucial to pay close attention to the signs and perform the operations on both sides of the equation consistently. Verifying your solution by substituting it back into the original equation is also an excellent way to catch any errors.

6. Tips and Tricks for Solving Equations

To become proficient in solving equations, here are some tips and tricks that can be helpful:

  • Simplify both sides first: Before attempting to isolate the variable, simplify each side of the equation by combining like terms and performing any necessary operations.
  • Use inverse operations: To isolate the variable, apply inverse operations. For example, if a term is added, subtract it from both sides; if a term is multiplied, divide both sides by it.
  • Maintain balance: Always perform the same operation on both sides of the equation to maintain equality.
  • Check your work: After finding a solution, substitute it back into the original equation to verify that it is correct.
  • Practice regularly: The more you practice, the more comfortable and confident you will become in solving equations.

7. Real-World Applications of Solving Equations

Solving equations is not just a mathematical exercise; it has real-world applications in various fields. From physics and engineering to economics and finance, the ability to solve equations is crucial for modeling and understanding real-world phenomena. For example, in physics, equations are used to describe motion, forces, and energy. In finance, equations are used to calculate interest rates, loan payments, and investment returns.

Understanding how to solve equations can also help in everyday situations, such as budgeting, cooking, and planning trips. By mastering this skill, you can make informed decisions and solve problems effectively in various aspects of your life. The ability to translate real-world problems into mathematical equations and solve them is a valuable asset in many professions and daily activities.

8. Conclusion

In conclusion, we have thoroughly explored the process of solving the equation 10 = 2 - 4(ax - 3) for x. By following a step-by-step approach, we successfully isolated x and found that x = 1/a. We also verified our solution and analyzed the answer choices to confirm its accuracy. Furthermore, we discussed common mistakes to avoid, helpful tips and tricks, and the real-world applications of solving equations.

Mastering the skill of solving equations is essential for success in mathematics and beyond. By understanding the fundamental principles and practicing regularly, you can develop your problem-solving abilities and tackle more complex mathematical challenges with confidence. Remember to always approach each equation methodically, paying close attention to detail and verifying your solutions to ensure accuracy.