Solving For X In (3x - 6)/3 = (7x - 3)/6 A Step-by-Step Guide

Linear equations are fundamental in mathematics, serving as the building blocks for more complex concepts. Mastering the techniques to solve these equations is crucial for success in algebra and beyond. This article delves into the step-by-step process of solving the equation (3x - 6)/3 = (7x - 3)/6, providing a clear understanding of the underlying principles and methods involved. We will explore how to manipulate equations, isolate variables, and ultimately arrive at the correct solution. Whether you're a student grappling with algebra or simply looking to refresh your math skills, this guide will equip you with the knowledge and confidence to tackle similar problems with ease.

Understanding the Equation

At the heart of our problem lies the equation (3x - 6)/3 = (7x - 3)/6. This equation represents a relationship between two algebraic expressions, each involving the variable 'x'. To solve for x, our primary goal is to isolate 'x' on one side of the equation. This involves performing a series of algebraic manipulations that maintain the equality of both sides. Before we dive into the steps, it's crucial to understand the properties of equality. These properties dictate that performing the same operation on both sides of an equation does not alter its balance. For example, adding, subtracting, multiplying, or dividing both sides by the same non-zero number preserves the equality. With this foundation, we can confidently embark on the journey of solving for 'x' in this equation.

Step-by-Step Solution

Let's embark on a step-by-step journey to unravel the value of 'x' in the equation (3x - 6)/3 = (7x - 3)/6. Our initial goal is to eliminate the fractions, which often complicate the solving process. To achieve this, we'll employ a technique known as cross-multiplication. This involves multiplying the numerator of the left-hand side by the denominator of the right-hand side, and vice versa. Applying this to our equation, we multiply (3x - 6) by 6 and (7x - 3) by 3. This yields the equation 6 * (3x - 6) = 3 * (7x - 3), which is free of fractions and sets the stage for further simplification.

Eliminating Fractions

The initial step in solving the equation (3x - 6)/3 = (7x - 3)/6 is to eliminate the fractions. This simplifies the equation and makes it easier to work with. We can achieve this by multiplying both sides of the equation by the least common multiple (LCM) of the denominators, which in this case is 6. Multiplying both sides by 6, we get: 6 * [(3x - 6)/3] = 6 * [(7x - 3)/6]. This simplifies to 2 * (3x - 6) = (7x - 3). This step effectively removes the fractions, paving the way for further simplification and ultimately solving for 'x'.

Applying the Distributive Property

Now that we've successfully eliminated the fractions, the next crucial step is to apply the distributive property. This property states that a(b + c) = ab + ac, and it's essential for simplifying expressions involving parentheses. In our equation, 2 * (3x - 6) = (7x - 3), we need to distribute the 2 on the left-hand side. Multiplying 2 by both 3x and -6, we obtain 6x - 12. Similarly, on the right-hand side, we have (7x - 3), which remains unchanged for now. Our equation now transforms to 6x - 12 = 7x - 3. By carefully applying the distributive property, we've expanded the equation and prepared it for the next phase of isolating 'x'.

Isolating the Variable

With the distributive property applied, our equation now reads 6x - 12 = 7x - 3. The next strategic move is to isolate the variable 'x' on one side of the equation. This involves gathering all terms containing 'x' on one side and all constant terms on the other. To achieve this, we can subtract 6x from both sides of the equation. This eliminates the 'x' term from the left side, resulting in -12 = 7x - 6x - 3, which simplifies to -12 = x - 3. By performing this subtraction, we've made significant progress in isolating 'x' and bringing us closer to the solution.

Solving for x

Having successfully isolated the variable terms on one side, our equation stands at -12 = x - 3. The final step in solving for 'x' is to isolate it completely. To achieve this, we need to eliminate the constant term, -3, from the right-hand side. We can accomplish this by adding 3 to both sides of the equation. This operation maintains the balance of the equation while effectively isolating 'x'. Adding 3 to both sides gives us -12 + 3 = x - 3 + 3, which simplifies to -9 = x. Therefore, the solution to the equation is x = -9. We have now successfully navigated the steps to solve for 'x' and arrived at the answer.

Verification

Before declaring victory, it's always prudent to verify our solution. This ensures that our calculated value of x indeed satisfies the original equation. To verify, we substitute x = -9 back into the equation (3x - 6)/3 = (7x - 3)/6. Plugging in -9 for x, we get [3*(-9) - 6]/3 = [7*(-9) - 3]/6. Simplifying the left-hand side, we have (-27 - 6)/3 = -33/3 = -11. On the right-hand side, we get (-63 - 3)/6 = -66/6 = -11. Since both sides of the equation evaluate to -11, our solution x = -9 is indeed correct. This verification step solidifies our confidence in the accuracy of our solution and underscores the importance of checking our work.

The Correct Answer

Through our step-by-step journey, we've successfully navigated the equation (3x - 6)/3 = (7x - 3)/6 and arrived at the solution x = -9. We meticulously eliminated fractions, applied the distributive property, isolated the variable, and verified our answer to ensure its accuracy. Therefore, the correct answer is B. -9.

Conclusion

In conclusion, solving linear equations is a fundamental skill in mathematics, and mastering it opens doors to more advanced concepts. This article has provided a comprehensive guide to solving the equation (3x - 6)/3 = (7x - 3)/6, demonstrating the step-by-step process of eliminating fractions, applying the distributive property, isolating variables, and verifying the solution. By understanding these principles and practicing diligently, you can confidently tackle a wide range of linear equations. Remember, the key to success in mathematics lies in a solid grasp of the fundamentals and a willingness to practice and apply what you've learned. So, embrace the challenge, sharpen your skills, and embark on your mathematical journey with confidence.

Keywords: Linear equations, solve for x, algebra, equation solving, distributive property, isolating variables, verification, mathematics, step-by-step solution, equation manipulation, solving techniques, algebraic expressions, equation balance, least common multiple, cross-multiplication.