Solve The Equation -x + 8 + 3x = X - 6.
Introduction
In this article, we will walk through the step-by-step solution of the equation -x + 8 + 3x = x - 6. Understanding how to solve linear equations is a fundamental skill in algebra, and this example will provide a clear and concise method to tackle similar problems. Our main focus will be on combining like terms, isolating the variable, and verifying the solution. This comprehensive guide aims to help students and anyone interested in mathematics to strengthen their algebraic skills. Let's dive in!
Problem Statement
The equation we need to solve is: -x + 8 + 3x = x - 6. This is a linear equation, meaning the highest power of the variable x is 1. Our goal is to find the value of x that makes this equation true. To do this, we will simplify the equation by combining like terms and isolating x on one side. This process involves several key algebraic manipulations that we will explore in detail.
Step-by-Step Solution
Step 1: Combine Like Terms on the Left Side
In this first step, we focus on simplifying the left side of the equation. We identify the like terms, which are terms that have the same variable raised to the same power. In our equation, -x and 3x are like terms. To combine them, we add their coefficients: -1 + 3 = 2. So, -x + 3x simplifies to 2x. The equation now looks like this:
2x + 8 = x - 6
Combining like terms is a crucial step because it reduces the complexity of the equation, making it easier to handle. By grouping similar terms together, we make the equation more manageable and set the stage for the next steps in the solution process. This step ensures that we're working with the simplest form of the equation before we move on to further manipulations.
Step 2: Move the Variable Terms to One Side
Next, we want to group all the terms containing x on one side of the equation. To do this, we subtract x from both sides. This maintains the equation's balance, a fundamental principle in algebra. Subtracting x from both sides gives us:
2x + 8 - x = x - 6 - x
Simplifying this, we get:
x + 8 = -6
The purpose of this step is to isolate the variable x on one side, which brings us closer to finding its value. By subtracting x from both sides, we eliminate x from the right side and consolidate all variable terms on the left. This is a standard technique in solving equations and is essential for efficiently finding the solution.
Step 3: Isolate the Variable
Now, we need to isolate x completely. Currently, we have x with an added constant term (+8). To isolate x, we subtract 8 from both sides of the equation. Again, this maintains the balance of the equation:
x + 8 - 8 = -6 - 8
This simplifies to:
x = -14
By subtracting 8 from both sides, we effectively remove the constant term from the left side, leaving x alone. This step is the culmination of our efforts to isolate the variable and directly reveals the value of x. The result, x = -14, is our solution, but we will verify it in the next step to ensure its correctness.
Step 4: Verify the Solution
To ensure our solution is correct, we substitute x = -14 back into the original equation: -x + 8 + 3x = x - 6.
Substituting x = -14 gives us:
-(-14) + 8 + 3(-14) = -14 - 6
Simplifying the left side:
14 + 8 - 42 = -20
22 - 42 = -20
-20 = -20
And the right side:
-14 - 6 = -20
Since both sides of the equation are equal (-20 = -20), our solution x = -14 is correct. Verifying the solution is a crucial step in the problem-solving process. It provides a check against errors made during the simplification and isolation steps. By substituting the solution back into the original equation and confirming that both sides are equal, we gain confidence in the accuracy of our answer.
Final Answer
The solution to the equation -x + 8 + 3x = x - 6 is x = -14. We arrived at this solution by systematically combining like terms, moving variable terms to one side, isolating the variable, and finally verifying our answer. This process demonstrates the fundamental techniques used in solving linear equations, which are applicable to a wide range of algebraic problems. Understanding these steps is essential for anyone studying mathematics and provides a solid foundation for more advanced topics.
Conclusion
In summary, we have successfully solved the equation -x + 8 + 3x = x - 6 by following a clear, step-by-step method. We combined like terms, isolated the variable, and verified our solution. The final answer is x = -14. Mastering these algebraic techniques is crucial for mathematical proficiency. We hope this detailed explanation has helped you understand the process better and has equipped you with the skills to tackle similar problems in the future. Remember, practice is key to mastering mathematics, so keep solving equations and expanding your knowledge!