Solving For X: A Step-by-Step Guide To -x + 3/7 = 2x - 25/7

Are you grappling with the equation -x + 3/7 = 2x - 25/7 and seeking a clear, step-by-step solution? You've come to the right place! This comprehensive guide will walk you through the process of isolating x and determining its value. Understanding how to solve linear equations like this is a fundamental skill in algebra, opening doors to more complex mathematical concepts. So, let's dive in and conquer this equation together!

Understanding the Basics of Algebraic Equations

Before we jump into the solution, let's briefly touch on the basics. An algebraic equation is a mathematical statement that asserts the equality of two expressions. These expressions typically involve variables (like our x), constants (numbers), and mathematical operations (addition, subtraction, multiplication, division). The goal of solving an equation is to find the value(s) of the variable(s) that make the equation true. In simpler terms, we want to find the number that x represents in this specific case. We achieve this by manipulating the equation while maintaining the equality, a principle that's key to accurate problem-solving in algebra.

When we talk about "solving for x," we're essentially trying to isolate x on one side of the equation. This means performing operations on both sides of the equation until we have x all by itself on one side and a numerical value on the other. Remember, whatever operation we perform on one side, we must perform on the other to maintain the balance and keep the equation valid. This principle of maintaining balance is the cornerstone of solving algebraic equations. By understanding this fundamental concept, we can approach any linear equation with confidence, knowing that we have the tools to manipulate it and arrive at the correct solution.

Step 1: Combining Like Terms

The initial step in solving the equation -x + 3/7 = 2x - 25/7 involves gathering similar terms. Our main objective here is to consolidate the x terms on one side of the equation and the constant terms on the other. This strategic rearrangement simplifies the equation and brings us closer to isolating x. To achieve this, we'll perform specific operations on both sides of the equation, ensuring that the equality remains intact.

Let's begin by focusing on the x terms. We have '-x' on the left side and '2x' on the right. To bring these terms together, we can add 'x' to both sides of the equation. This action eliminates the '-x' term on the left and adds it to the '2x' term on the right. The equation then transforms into 3/7 = 2x + x - 25/7. Simplifying the right side, we get 3/7 = 3x - 25/7. This step is crucial as it centralizes the variable we're solving for, making the subsequent steps more straightforward.

Next, we'll deal with the constant terms, which are the numbers without any variables attached. We have '3/7' on the left side and '-25/7' on the right. To group these constants, we add '25/7' to both sides of the equation. This eliminates the '-25/7' on the right and adds it to the '3/7' on the left. The equation now looks like 3/7 + 25/7 = 3x. Combining the fractions on the left, we have 28/7 = 3x. This simplification is a key step in isolating x, bringing us closer to our final solution. By systematically combining like terms, we've streamlined the equation, making it easier to solve.

Step 2: Simplifying the Equation

Following the combination of like terms, the next crucial step is simplifying the equation. In our case, we've arrived at 28/7 = 3x. This equation, while more concise than the original, still needs further refinement to isolate x. The simplification process involves reducing fractions and performing any possible arithmetic operations to make the equation as straightforward as possible. By simplifying, we not only make the equation easier to work with but also reduce the chances of making errors in the subsequent steps.

The left side of the equation, 28/7, is a fraction that can be simplified. Both the numerator (28) and the denominator (7) are divisible by 7. Dividing both 28 and 7 by 7, we get 4. Therefore, 28/7 simplifies to 4. Replacing 28/7 with 4, our equation now reads 4 = 3x. This simplification is a significant step forward, as it replaces a fraction with a whole number, making the equation much cleaner and easier to handle.

The simplified equation, 4 = 3x, is now in a form where the path to isolating x is much clearer. We have a simple equation where a constant (4) is equal to another constant (3) multiplied by x. This form is ideal for the final step of solving for x, which involves dividing both sides of the equation by the coefficient of x. By simplifying the equation, we've set the stage for a direct and efficient solution, demonstrating the power of simplification in algebraic problem-solving.

Step 3: Isolating x

Having simplified the equation to 4 = 3x, the final step in solving for x is to isolate the variable. Isolating x means getting x by itself on one side of the equation. This is achieved by performing the inverse operation of whatever is being done to x. In our equation, x is being multiplied by 3. The inverse operation of multiplication is division, so we need to divide both sides of the equation by 3.

Dividing both sides of the equation 4 = 3x by 3, we get 4/3 = (3x)/3. On the right side, the 3 in the numerator and the 3 in the denominator cancel each other out, leaving us with just x. This cancellation is a direct result of performing the inverse operation, which is a fundamental technique in solving algebraic equations. After the division and cancellation, our equation transforms into 4/3 = x. This is the solution we've been seeking, where x is now isolated and its value is clearly stated.

The solution x = 4/3 means that the value of x that makes the original equation true is 4/3. This result can be expressed as an improper fraction (4/3) or as a mixed number (1 1/3). Both forms represent the same value and are acceptable solutions. By isolating x, we've successfully solved the equation, demonstrating the power of inverse operations in algebraic problem-solving. This final step underscores the importance of understanding and applying inverse operations to isolate variables and find solutions.

Conclusion

In conclusion, solving the equation -x + 3/7 = 2x - 25/7 involves a series of strategic steps aimed at isolating x. We began by combining like terms, a process that simplifies the equation by grouping similar elements together. This was followed by simplifying the equation further, which involved reducing fractions and performing arithmetic operations to make the equation more manageable. Finally, we isolated x by performing the inverse operation, which in this case was division. The solution we arrived at is x = 4/3, a value that satisfies the original equation.

This step-by-step approach is a cornerstone of algebra and can be applied to a wide range of linear equations. The key to success lies in understanding the principles of equality and inverse operations. By maintaining balance in the equation and systematically isolating the variable, you can confidently solve for x in various algebraic scenarios. Remember, each step builds upon the previous one, leading you closer to the solution. With practice and a clear understanding of these fundamental concepts, you'll be well-equipped to tackle more complex algebraic challenges.

For further exploration and practice with algebraic equations, you might find valuable resources and exercises on websites like Khan Academy Algebra. Happy solving!