Solve The Equation 5x - 17 = -x + 7 For X.
In this article, we will solve the algebraic equation 5x - 17 = -x + 7 step-by-step to find the value of x. This problem falls under the category of basic algebra, which is a fundamental concept in mathematics. We will explore the methods used to isolate the variable x and arrive at the solution. Understanding how to solve such equations is crucial for tackling more complex mathematical problems. The ability to manipulate equations and solve for unknowns is a core skill in algebra and is widely applied in various fields such as physics, engineering, and economics. This article aims to provide a clear and concise explanation of the solution process, making it easy for readers to follow and understand. Mastering these basic algebraic techniques will build a strong foundation for further mathematical studies and practical applications. Let's dive into the detailed solution and unravel the value of x in this equation. By working through this example, you'll gain confidence in solving similar algebraic problems and strengthen your problem-solving abilities. This foundational understanding is essential for advancing in mathematics and related disciplines. We will break down each step, ensuring clarity and reinforcing the underlying principles of algebraic manipulation. So, let's embark on this mathematical journey and discover the solution together, enhancing your algebraic skills along the way.
Understanding the Equation
The equation we need to solve is 5x - 17 = -x + 7. This is a linear equation in one variable, x. To find the value of x, our goal is to isolate x on one side of the equation. This involves using algebraic operations to manipulate the equation while maintaining its balance. The fundamental principle is that whatever operation you perform on one side of the equation, you must perform the same operation on the other side to keep the equation true. We'll start by gathering the terms with x on one side and the constant terms on the other side. This is a common strategy in solving linear equations and makes the subsequent steps clearer. Understanding the structure of the equation and the goal of isolating the variable is the first crucial step in solving any algebraic problem. By carefully applying algebraic principles, we can systematically simplify the equation and arrive at the solution for x. This process demonstrates the power of algebraic manipulation in unraveling the unknowns in an equation. Let's proceed with the next steps to see how we can isolate x and determine its value. Remember, each step is a logical progression that brings us closer to the solution.
Step-by-Step Solution
Step 1: Add x to both sides of the equation.
To begin isolating x, we need to eliminate the -x term on the right side. We can achieve this by adding x to both sides of the equation:
(5x - 17) + x = (-x + 7) + x
This simplifies to:
6x - 17 = 7
Adding x to both sides maintains the equality and moves us closer to isolating x. This is a fundamental step in solving linear equations, and it's crucial to ensure that the equation remains balanced. By adding x to both sides, we've successfully moved all the terms containing x to the left side of the equation. This is a common technique in algebra and helps to simplify the equation. The next step will involve isolating x further by dealing with the constant term on the left side.
Step 2: Add 17 to both sides of the equation.
Now, we need to isolate the term with x (6x) by eliminating the constant term -17 on the left side. We can do this by adding 17 to both sides of the equation:
(6x - 17) + 17 = 7 + 17
This simplifies to:
6x = 24
Adding 17 to both sides isolates the term with x on the left side and combines the constants on the right side. This is another crucial step in solving for x. By performing this operation, we've made the equation even simpler, bringing us closer to the final solution. The next step will be to divide both sides by the coefficient of x to finally solve for x.
Step 3: Divide both sides by 6.
To finally isolate x, we need to divide both sides of the equation by the coefficient of x, which is 6:
6x / 6 = 24 / 6
This simplifies to:
x = 4
Dividing both sides by 6 gives us the value of x. This is the final step in solving the equation. We have successfully isolated x and found its value. The solution x = 4 satisfies the original equation, making it the correct answer.
Final Answer
Therefore, if 5x - 17 = -x + 7, then x = 4. The correct answer is C. 4.
We have systematically solved the equation by applying algebraic principles, ensuring that each step maintains the balance of the equation. This step-by-step approach is a valuable technique for solving various algebraic problems. The key is to understand the goal of isolating the variable and applying appropriate operations to achieve this. By mastering these techniques, you'll be well-equipped to tackle more complex mathematical challenges. The solution x = 4 is the only value that satisfies the given equation, making it the unique solution. This process demonstrates the power of algebraic manipulation in finding unknown values and solving equations.
Conclusion
In conclusion, we successfully solved the equation 5x - 17 = -x + 7 by systematically isolating x using algebraic operations. We added x to both sides, then added 17 to both sides, and finally divided both sides by 6 to find that x = 4. This demonstrates a fundamental method for solving linear equations. The step-by-step approach ensures that the equation remains balanced throughout the process, leading to the correct solution. Understanding these techniques is crucial for success in algebra and other areas of mathematics. By mastering these concepts, you'll be able to tackle a wide range of problems involving equations and variables. The ability to manipulate equations and solve for unknowns is a valuable skill that extends beyond the classroom and into various practical applications. Remember, practice is key to mastering these techniques, so continue to work through examples and build your confidence in solving algebraic equations. The solution x = 4 is the final answer and represents the value that makes the original equation true. This process highlights the importance of precision and attention to detail in algebraic problem-solving.