Solving (2/3)x + 1 = (1/6)x - 7 A Step-by-Step Guide
In this article, we will delve into the process of solving the equation (2/3)x + 1 = (1/6)x - 7. This is a linear equation, and our goal is to isolate the variable 'x' to find its value. We will walk through each step methodically, ensuring a clear understanding of the algebraic manipulations involved. This is a fundamental skill in algebra, and mastering it will pave the way for tackling more complex equations in the future.
1. Clearing the Fractions: The First Crucial Step
The presence of fractions in an equation can often make it appear more daunting than it actually is. Therefore, our first step in solving this equation is to eliminate these fractions. To do this, we need to find the least common multiple (LCM) of the denominators, which in this case are 3 and 6. The LCM of 3 and 6 is 6. Multiplying both sides of the equation by 6 will clear the fractions and simplify the equation significantly.
When we multiply both sides of the equation (2/3)x + 1 = (1/6)x - 7 by 6, we get:
6 * [(2/3)x + 1] = 6 * [(1/6)x - 7]
Distributing the 6 on both sides, we have:
6 * (2/3)x + 6 * 1 = 6 * (1/6)x - 6 * 7
Simplifying each term:
(12/3)x + 6 = (6/6)x - 42
Which further simplifies to:
4x + 6 = x - 42
Now, the equation looks much cleaner and easier to work with. We have successfully eliminated the fractions, which is a crucial step in solving many algebraic equations. This step reduces the risk of errors and makes the subsequent steps more straightforward. The equation 4x + 6 = x - 42 is now free of fractions, allowing us to focus on isolating the variable 'x'.
2. Isolating the Variable 'x': Bringing Like Terms Together
Now that we have cleared the fractions, our next objective is to isolate the variable 'x'. This means we need to get all the terms containing 'x' on one side of the equation and all the constant terms on the other side. To achieve this, we can use the properties of equality, which allow us to add or subtract the same value from both sides of the equation without changing its balance. In the equation 4x + 6 = x - 42, we have 'x' terms on both sides, so we need to consolidate them.
Let's start by subtracting 'x' from both sides of the equation:
4x + 6 - x = x - 42 - x
This simplifies to:
3x + 6 = -42
Now, we have all the 'x' terms on the left side of the equation. Next, we need to move the constant term (+6) to the right side. To do this, we subtract 6 from both sides of the equation:
3x + 6 - 6 = -42 - 6
This simplifies to:
3x = -48
We have now successfully isolated the 'x' term on the left side and the constant term on the right side. The equation 3x = -48 is much simpler to solve than the original equation. By bringing like terms together, we have made significant progress towards finding the value of 'x'.
3. Solving for 'x': The Final Calculation
We've reached the final stage of solving the equation. We have successfully simplified the equation to 3x = -48. Now, to find the value of 'x', we need to isolate 'x' completely. Since 'x' is being multiplied by 3, we can use the inverse operation, which is division, to isolate 'x'. We will divide both sides of the equation by 3.
Dividing both sides of the equation 3x = -48 by 3, we get:
(3x) / 3 = -48 / 3
This simplifies to:
x = -16
Therefore, the solution to the equation (2/3)x + 1 = (1/6)x - 7 is x = -16. This is the value of 'x' that makes the equation true. We have successfully solved for 'x' by performing a series of algebraic manipulations, including clearing fractions, combining like terms, and using inverse operations.
4. Verifying the Solution: Ensuring Accuracy
It's always a good practice to verify your solution, especially in mathematics. This step ensures that the value you found for 'x' is indeed correct and satisfies the original equation. To verify our solution, we will substitute x = -16 back into the original equation (2/3)x + 1 = (1/6)x - 7 and see if both sides of the equation are equal.
Substituting x = -16 into the equation, we get:
(2/3)(-16) + 1 = (1/6)(-16) - 7
Now, we simplify each side of the equation separately.
Left side:
(2/3)(-16) + 1 = -32/3 + 1
To add these terms, we need a common denominator, which is 3:
-32/3 + 3/3 = -29/3
Right side:
(1/6)(-16) - 7 = -16/6 - 7
We can simplify -16/6 by dividing both the numerator and denominator by 2:
-8/3 - 7
Again, we need a common denominator to subtract these terms. The common denominator is 3:
-8/3 - 21/3 = -29/3
Comparing both sides:
-29/3 = -29/3
Since both sides of the equation are equal when x = -16, we have verified that our solution is correct. This step provides confidence in our answer and demonstrates a thorough understanding of the problem-solving process. Verifying the solution is a critical step in any mathematical problem, as it helps to catch errors and ensure accuracy.
Conclusion: The Solution and Key Takeaways
In conclusion, we have successfully solved the equation (2/3)x + 1 = (1/6)x - 7 and found that x = -16. We achieved this by following a systematic approach: first, we cleared the fractions by multiplying both sides of the equation by the least common multiple of the denominators. Then, we isolated the variable 'x' by bringing like terms together, using the properties of equality. Finally, we solved for 'x' by using the inverse operation of division. To ensure the accuracy of our solution, we verified it by substituting the value of 'x' back into the original equation and confirming that both sides were equal.
This problem demonstrates several important concepts in algebra. Clearing fractions is a crucial technique for simplifying equations and making them easier to solve. Isolating the variable is the core goal in solving any equation, and it involves using the properties of equality to manipulate the equation while maintaining its balance. Verification is a vital step in the problem-solving process, as it helps to identify and correct any errors.
Understanding and mastering these techniques will be invaluable as you tackle more complex algebraic problems in the future. Remember to approach each equation systematically, break it down into smaller steps, and always verify your solution. With practice and a solid understanding of these fundamental concepts, you will become proficient in solving a wide range of algebraic equations.
Therefore, the correct answer is:
A. x = -16