Solving (3x - 6)/3 = (7x - 3)/6 A Step-by-Step Guide
Step 1: Eliminating the Fractions
To solve the equation, our first goal is to eliminate the fractions. This makes the equation easier to work with. Eliminating fractions in the equation (3x - 6)/3 = (7x - 3)/6 can be achieved by multiplying both sides of the equation by the least common multiple (LCM) of the denominators. The denominators in our equation are 3 and 6. The least common multiple of 3 and 6 is 6. Therefore, we multiply both sides of the equation by 6. This step simplifies the equation and makes it easier to solve for 'x'. Multiplying by the LCM is a standard technique in algebra to clear fractions from an equation, making it more manageable.
So, we multiply both sides of the equation by 6:
6 * [(3x - 6)/3] = 6 * [(7x - 3)/6]
This simplifies to:
2(3x - 6) = 7x - 3
Multiplying both sides by 6 is crucial in clearing the fractions. By doing this, we transform the original equation into a more straightforward linear equation. This step is based on the fundamental algebraic principle that multiplying both sides of an equation by the same non-zero number maintains the equality. The result, 2(3x - 6) = 7x - 3, is now free of fractions and ready for further simplification.
Step 2: Distributing and Simplifying
Now, we need to distribute the 2 on the left side of the equation. Distributing the 2 in the equation 2(3x - 6) = 7x - 3 involves multiplying the 2 by each term inside the parentheses. This is an application of the distributive property, which is a fundamental concept in algebra. The distributive property states that a(b + c) = ab + ac. Applying this property helps us to expand the expression and combine like terms, bringing us closer to isolating 'x'. Distributing correctly is essential for accurate equation solving.
2(3x - 6) becomes 6x - 12.
So, our equation now looks like this:
6x - 12 = 7x - 3
The next step is to simplify the equation by combining like terms. Simplifying the equation 6x - 12 = 7x - 3 involves rearranging the terms so that 'x' terms are on one side and constant terms are on the other. This is a crucial step in isolating 'x' and finding its value. By simplifying, we reduce the complexity of the equation and make it easier to solve. This process typically involves adding or subtracting terms from both sides of the equation to maintain balance.
Step 3: Isolating the Variable
To isolate the variable 'x', we need to get all the 'x' terms on one side of the equation and the constants on the other side. Isolating 'x' in the equation 6x - 12 = 7x - 3 is a key step in solving for 'x'. This involves performing operations on both sides of the equation to group the 'x' terms and the constant terms separately. The goal is to have 'x' on one side and a numerical value on the other, which will give us the solution. This step often requires adding or subtracting terms from both sides to maintain equality.
Let's subtract 6x from both sides:
6x - 12 - 6x = 7x - 3 - 6x
This simplifies to:
-12 = x - 3
Now, we add 3 to both sides:
-12 + 3 = x - 3 + 3
This simplifies to:
-9 = x
So, x = -9.
Subtracting 6x from both sides is a strategic move to consolidate the 'x' terms on one side of the equation. This step helps to simplify the equation and move closer to isolating 'x'. By subtracting 6x, we maintain the balance of the equation while reducing the complexity. This technique is fundamental in algebraic manipulation and is crucial for solving linear equations efficiently.
Step 4: Verifying the Solution
It's always a good idea to verify our solution by plugging it back into the original equation. Verifying the solution x = -9 involves substituting -9 for 'x' in the original equation (3x - 6)/3 = (7x - 3)/6. This step is crucial to ensure that our solution is correct and satisfies the equation. If both sides of the equation are equal after the substitution, then our solution is verified. This process helps to catch any errors made during the solving process and provides confidence in the final answer.
Original equation:
(3x - 6)/3 = (7x - 3)/6
Substitute x = -9:
[3(-9) - 6]/3 = [7(-9) - 3]/6
Simplify the left side:
(-27 - 6)/3 = -33/3 = -11
Simplify the right side:
(-63 - 3)/6 = -66/6 = -11
Since both sides are equal (-11 = -11), our solution x = -9 is correct.
Substituting x = -9 into the original equation is a critical step to confirm the accuracy of our solution. This process involves replacing every instance of 'x' in the equation with -9 and then simplifying both sides. If the left-hand side and the right-hand side of the equation are equal after simplification, then the solution is verified. This step provides a foolproof way to ensure that the calculated value of 'x' is indeed the correct solution to the equation.
Final Answer
The value of x that makes the equation true is -9.
Therefore, the correct answer is B. -9. The final answer, x = -9, is the value that satisfies the original equation (3x - 6)/3 = (7x - 3)/6. This solution was obtained by following a systematic approach, including eliminating fractions, distributing terms, isolating the variable, and verifying the solution. Understanding and applying these steps is essential for solving linear equations accurately. The correct answer demonstrates a strong understanding of algebraic principles and problem-solving skills.
Conclusion
Solving the equation (3x - 6)/3 = (7x - 3)/6 involves several key steps, including eliminating fractions, distributing, isolating the variable, and verifying the solution. Understanding each step is crucial for mastering algebraic equations. By following this guide, you can confidently solve similar problems and improve your mathematical skills. Remember, solving equations like (3x - 6)/3 = (7x - 3)/6 is a fundamental skill in mathematics. By understanding the steps involved—clearing fractions, distributing, isolating the variable, and verifying the solution—you can tackle a wide range of algebraic problems. Practice and a systematic approach are key to mastering these techniques. This guide provides a comprehensive walkthrough, ensuring you grasp each concept thoroughly and can apply it effectively.
