Solving For X In The Equation 4(7x - 3) - 6 = 6 - 4(3x - 7)

JU07/24/2025 08, 2025 by THE IDEN 60 views

Solve for x in the Equation 4(7x - 3) - 6 = 6 - 4(3x - 7)

Introduction

In this article, we will delve into the process of solving for x in the linear equation 4(7x - 3) - 6 = 6 - 4(3x - 7). This type of problem is a fundamental concept in algebra, and mastering it is crucial for tackling more complex mathematical challenges. We'll break down the solution step by step, ensuring a clear understanding of each operation involved. Linear equations are ubiquitous in various fields, from engineering and physics to economics and computer science, making the ability to solve them an invaluable skill. Our aim is not just to provide the answer, but to equip you with a methodology that can be applied to a wide range of similar problems. We will explore the underlying principles of equation manipulation, such as the distributive property, combining like terms, and maintaining balance by performing the same operations on both sides. This comprehensive approach will empower you to confidently tackle any linear equation that comes your way. Through careful explanation and methodical steps, we will demystify the process of solving for x, making it accessible and understandable for learners of all levels. Whether you're a student preparing for an exam or simply looking to refresh your algebraic skills, this article will serve as a valuable resource. So, let's embark on this mathematical journey and unravel the solution to this equation together!

Detailed Solution

The primary goal in solving an algebraic equation is to isolate the variable, which in this case is x. This involves performing a series of operations on both sides of the equation to maintain equality while simplifying the expression. Our equation is 4(7x - 3) - 6 = 6 - 4(3x - 7). The first step is to apply the distributive property, which states that a(b + c) = ab + ac. This will help us eliminate the parentheses and simplify both sides of the equation. By distributing the 4 on the left side and the -4 on the right side, we get: 28x - 12 - 6 = 6 - 12x + 28. Next, we need to combine like terms on each side of the equation. Like terms are terms that have the same variable raised to the same power, or constants. On the left side, we combine -12 and -6 to get -18, and on the right side, we combine 6 and 28 to get 34. This simplifies our equation to: 28x - 18 = 34 - 12x. Now, we want to get all the x terms on one side of the equation and the constants on the other side. To do this, we add 12x to both sides of the equation to eliminate the x term on the right side: 28x + 12x - 18 = 34 - 12x + 12x, which simplifies to 40x - 18 = 34. Next, we add 18 to both sides of the equation to isolate the x term: 40x - 18 + 18 = 34 + 18, which simplifies to 40x = 52. Finally, to solve for x, we divide both sides of the equation by 40: 40x / 40 = 52 / 40, which gives us x = 52 / 40. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 4. This gives us the final solution: x = 13 / 10. Thus, the value of x that satisfies the original equation is 13 / 10, or 1.3 in decimal form. This detailed walkthrough provides a clear and concise method for solving linear equations, emphasizing the importance of each step and the underlying mathematical principles.

Step-by-Step Breakdown

To further clarify the process of solving the equation 4(7x - 3) - 6 = 6 - 4(3x - 7), let's break down each step individually. This step-by-step approach will not only aid in understanding the solution to this specific problem but also provide a framework for tackling similar equations in the future. Each step is crucial and builds upon the previous one, ensuring the equation remains balanced and the solution accurate. Step 1: Apply the Distributive Property. The distributive property is a fundamental concept in algebra that allows us to multiply a single term by multiple terms within parentheses. In our equation, we have 4(7x - 3) and -4(3x - 7). Applying the distributive property to the left side, we multiply 4 by both 7x and -3, resulting in 28x - 12. Similarly, on the right side, we multiply -4 by both 3x and -7, resulting in -12x + 28. Our equation now looks like this: 28x - 12 - 6 = 6 - 12x + 28. Step 2: Combine Like Terms. Like terms are terms that have the same variable raised to the same power or constants. On the left side of the equation, we have two constant terms, -12 and -6, which we can combine to get -18. On the right side, we have the constants 6 and 28, which combine to give 34. The equation simplifies to 28x - 18 = 34 - 12x. This step streamlines the equation, making it easier to manage and solve. Step 3: Isolate the Variable Terms. Our goal is to get all the terms with x on one side of the equation and the constant terms on the other side. To do this, we can add 12x to both sides of the equation. This eliminates the -12x term on the right side and adds it to the left side. The equation becomes 28x + 12x - 18 = 34 - 12x + 12x, which simplifies to 40x - 18 = 34. Step 4: Isolate the Constant Terms. Now, we want to isolate the x term by moving the constant term to the other side of the equation. We can do this by adding 18 to both sides. This eliminates the -18 on the left side and adds it to the right side. The equation becomes 40x - 18 + 18 = 34 + 18, which simplifies to 40x = 52. Step 5: Solve for x. Finally, to solve for x, we need to divide both sides of the equation by the coefficient of x, which is 40. This gives us 40x / 40 = 52 / 40, which simplifies to x = 52 / 40. We can further simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 4. This gives us the final solution: x = 13 / 10. Each of these steps is crucial in solving the equation, and understanding the logic behind each step is essential for mastering algebraic problem-solving. By breaking down the process into manageable steps, we make the solution more accessible and easier to follow.

Verification

After finding a solution to an equation, it is crucial to verify its correctness. This step ensures that the value obtained for x indeed satisfies the original equation. Verification involves substituting the solution back into the original equation and checking if both sides of the equation are equal. This process not only confirms the accuracy of the solution but also reinforces the understanding of the equation and the steps taken to solve it. In our case, the original equation is 4(7x - 3) - 6 = 6 - 4(3x - 7), and our solution is x = 13 / 10. To verify, we substitute x = 13 / 10 into the original equation. Step 1: Substitute x = 13/10 into the equation. Substituting x = 13 / 10 into 4(7x - 3) - 6 = 6 - 4(3x - 7), we get: 4(7(13 / 10) - 3) - 6 = 6 - 4(3(13 / 10) - 7). Step 2: Simplify the expressions inside the parentheses. First, we simplify 7(13 / 10), which equals 91 / 10. So the left side becomes 4(91 / 10 - 3) - 6. Next, we simplify 3(13 / 10), which equals 39 / 10. So the right side becomes 6 - 4(39 / 10 - 7). Now we need to express the constants 3 and 7 as fractions with a denominator of 10. 3 becomes 30 / 10 and 7 becomes 70 / 10. The equation now looks like this: 4(91 / 10 - 30 / 10) - 6 = 6 - 4(39 / 10 - 70 / 10). Step 3: Perform the subtractions inside the parentheses. Subtracting the fractions, we get: 91 / 10 - 30 / 10 = 61 / 10 on the left side and 39 / 10 - 70 / 10 = -31 / 10 on the right side. The equation becomes: 4(61 / 10) - 6 = 6 - 4(-31 / 10). Step 4: Multiply. Multiplying 4 by 61 / 10 gives us 244 / 10, and multiplying -4 by -31 / 10 gives us 124 / 10. The equation becomes: 244 / 10 - 6 = 6 + 124 / 10. Step 5: Express the constants as fractions with a denominator of 10. We express 6 as a fraction with a denominator of 10, which is 60 / 10. The equation becomes: 244 / 10 - 60 / 10 = 60 / 10 + 124 / 10. Step 6: Perform the additions and subtractions. Subtracting 60 / 10 from 244 / 10 gives us 184 / 10 on the left side. Adding 60 / 10 and 124 / 10 gives us 184 / 10 on the right side. The equation simplifies to: 184 / 10 = 184 / 10. Since both sides of the equation are equal, our solution x = 13 / 10 is correct. This detailed verification process underscores the importance of double-checking solutions to ensure accuracy and a thorough understanding of the problem.

Conclusion

In conclusion, solving the linear equation 4(7x - 3) - 6 = 6 - 4(3x - 7) involves a series of algebraic manipulations aimed at isolating the variable x. We began by applying the distributive property to eliminate parentheses, then combined like terms to simplify the equation. The subsequent steps involved isolating the x terms and constants on opposite sides of the equation, ultimately leading to the solution x = 13 / 10. To ensure the accuracy of our solution, we performed a verification step, substituting x = 13 / 10 back into the original equation and confirming that both sides were indeed equal. This process underscores the importance of methodical problem-solving in mathematics. Each step, from applying the distributive property to combining like terms and isolating the variable, plays a crucial role in arriving at the correct answer. Moreover, the verification step highlights the significance of checking one's work to ensure accuracy. The skills and techniques demonstrated in this article are fundamental to algebra and are applicable to a wide range of mathematical problems. Mastering these concepts not only builds confidence in one's mathematical abilities but also lays a solid foundation for more advanced topics. The ability to solve linear equations is a valuable asset in various fields, from science and engineering to economics and finance. By understanding the underlying principles and practicing consistently, anyone can develop proficiency in solving these types of problems. The step-by-step approach outlined in this article serves as a template for tackling similar equations, emphasizing clarity, organization, and attention to detail. Ultimately, the journey of solving mathematical problems is not just about finding the answer but also about developing critical thinking skills and a deeper appreciation for the elegance and precision of mathematics. With practice and perseverance, anyone can unlock the power of algebra and confidently tackle any equation that comes their way.