Solving For X In The Equation -x + 3/7 = 2x - 25/7 A Step-by-Step Guide

Linear equations are fundamental in mathematics and are used extensively in various fields, from science and engineering to economics and finance. Understanding how to solve them is a crucial skill for anyone pursuing studies or careers involving quantitative analysis. In this article, we will delve into a step-by-step approach to solving a specific linear equation, highlighting the underlying principles and techniques involved. This will not only help you solve this particular problem but also equip you with the knowledge to tackle a wide range of similar equations. Our focus equation is $-x + \frac{3}{7} = 2x - \frac{25}{7}$, and we will explore the process of isolating the variable x to find its value. This journey will involve combining like terms, performing arithmetic operations on both sides of the equation, and simplifying the result to arrive at the correct solution. By the end of this guide, you'll have a clear understanding of how to solve linear equations and the confidence to apply these methods to other mathematical challenges. We will cover the importance of maintaining balance in an equation, the strategy of moving terms across the equals sign, and the ultimate goal of expressing x in terms of known constants. Let's begin our exploration of linear equations and unlock the mystery of solving for x.

Understanding the Equation: $-x + \frac{3}{7} = 2x - \frac{25}{7}$

Before diving into the solution, it's important to understand the structure of the equation $-x + \frac{3}{7} = 2x - \frac{25}{7}$. This equation is a linear equation in one variable, x. A linear equation is characterized by the fact that the highest power of the variable is 1. In simpler terms, x appears without any exponents (like $x^2$ or $x^3$) or other functions applied to it (like sin(x) or log(x)). This makes linear equations relatively straightforward to solve compared to more complex types of equations. The equation consists of two sides, the left-hand side (LHS) and the right-hand side (RHS), separated by an equals sign (=). The equals sign signifies that the expressions on both sides have the same value. Our goal is to find the value of x that makes this equality true. On the LHS, we have $-x + \frac{3}{7}$, which means we're starting with the negative of x and adding the fraction $\frac{3}{7}$. On the RHS, we have $2x - \frac{25}{7}$, which means we're taking two times x and subtracting the fraction $\frac{25}{7}$. The fractions $\frac{3}{7}$ and $\frac{25}{7}$ are constants, meaning their values are fixed and do not change. To solve for x, we need to isolate it on one side of the equation. This means manipulating the equation using algebraic operations until we have x by itself on one side and a constant value on the other side. This process involves moving terms around, combining like terms, and simplifying the equation. The key to solving any equation is to perform the same operations on both sides, ensuring that the equality is maintained. We'll see how this principle is applied as we move through the steps of solving for x in this equation.

Step 1: Combining Like Terms – Moving x Terms to One Side

The first step in solving the equation $-x + \frac{3}{7} = 2x - \frac{25}{7}$ is to gather all the terms containing x on one side of the equation. This process is known as combining like terms. Our goal is to isolate x, and bringing all the x terms together is a crucial step in achieving that. To do this, we can add x to both sides of the equation. This will eliminate the -x term on the left-hand side and move it to the right-hand side. Adding x to both sides maintains the balance of the equation, as we are performing the same operation on both sides. Here's how it looks mathematically:

$-x + \frac{3}{7} + x = 2x - \frac{25}{7} + x$

On the left-hand side, -x and +x cancel each other out, leaving us with just $\frac{3}{7}$. On the right-hand side, we have 2x + x, which simplifies to 3x. So, the equation now becomes:

$\frac{3}{7} = 3x - \frac{25}{7}$

This equation is simpler than the original one, as we have successfully moved all the x terms to the right-hand side. Now, we need to isolate the x term further by dealing with the constant term on the right-hand side. The next step will involve moving the constant term to the left-hand side, continuing our progress toward isolating x. This process of combining like terms is a fundamental technique in solving equations, allowing us to simplify the equation and make it easier to solve. By carefully performing the same operations on both sides, we ensure that the equation remains balanced and that we are moving closer to the correct solution.

Step 2: Isolating the x Term – Moving Constants to the Other Side

Having consolidated the x terms on one side, our next objective in solving $-x + \frac{3}{7} = 2x - \frac{25}{7}$ is to isolate the x term completely. From the previous step, we arrived at the equation $\frac{3}{7} = 3x - \frac{25}{7}$. Now, we need to move the constant term, $-\frac{25}{7}$, from the right-hand side to the left-hand side. To accomplish this, we add $\frac{25}{7}$ to both sides of the equation. This operation will eliminate the $-\frac{25}{7}$ term on the right-hand side and add its equivalent to the left-hand side. Remember, maintaining balance is key, so whatever we do to one side, we must do to the other. The equation after adding $\frac{25}{7}$ to both sides looks like this:

$\frac{3}{7} + \frac{25}{7} = 3x - \frac{25}{7} + \frac{25}{7}$

On the left-hand side, we have $\frac{3}{7} + \frac{25}{7}$, which are fractions with a common denominator. We can simply add the numerators: 3 + 25 = 28. So, the left-hand side becomes $\frac{28}{7}$. On the right-hand side, $-\frac{25}{7}$ and $+\frac{25}{7}$ cancel each other out, leaving us with just 3x. Now, our equation is:

$\frac{28}{7} = 3x$

We can simplify $\frac{28}{7}$ by dividing 28 by 7, which gives us 4. So, the equation further simplifies to:

$4 = 3x$

We are now very close to solving for x. The x term is almost isolated, with only the coefficient 3 remaining. The next step will involve eliminating this coefficient to find the value of x.

Step 3: Solving for x – Dividing to Isolate the Variable

We've made significant progress in solving the equation $-x + \frac{3}{7} = 2x - \frac{25}{7}$. After combining like terms and moving constants to one side, we arrived at the simplified equation $4 = 3x$. Our final step is to isolate x completely. Currently, x is being multiplied by 3. To undo this multiplication and solve for x, we need to perform the inverse operation, which is division. We will divide both sides of the equation by 3. This will eliminate the coefficient 3 from the right-hand side, leaving x by itself. Dividing both sides by 3 ensures that the equation remains balanced. Here's the mathematical representation of this step:

$\frac{4}{3} = \frac{3x}{3}$

On the left-hand side, we have $\frac{4}{3}$, which is a fraction that cannot be simplified further. On the right-hand side, we have $\frac{3x}{3}$. The 3 in the numerator and the 3 in the denominator cancel each other out, leaving us with just x. Therefore, the equation simplifies to:

$\frac{4}{3} = x$

This equation tells us that x is equal to $\frac{4}{3}$. We have successfully isolated x and found its value. This is the solution to the original equation. To summarize, we started with a linear equation, combined like terms, moved constants to one side, and finally, divided to isolate x. This step-by-step process is a powerful method for solving linear equations, and it can be applied to a wide variety of similar problems. The solution we found, $x = \frac{4}{3}$, corresponds to option C in the original question.

The Solution: x = 4/3 (Option C)

After meticulously following the steps of combining like terms, isolating the x term, and performing the necessary division, we have arrived at the solution for the equation $-x + \frac{3}{7} = 2x - \frac{25}{7}$. Our calculations have shown that the value of x that satisfies this equation is $\frac{4}{3}$. This solution corresponds to option C in the original multiple-choice question. To recap the process, we initially combined the x terms by adding x to both sides of the equation. This resulted in the equation $\frac{3}{7} = 3x - \frac{25}{7}$. Next, we isolated the x term by adding $\frac{25}{7}$ to both sides, leading to the equation $4 = 3x$. Finally, we solved for x by dividing both sides by 3, which gave us the solution $x = \frac{4}{3}$. This step-by-step approach demonstrates the systematic way in which linear equations can be solved. By carefully applying algebraic principles and maintaining balance in the equation, we were able to isolate the variable x and determine its value. The solution $x = \frac{4}{3}$ is not only the correct answer for this specific problem but also a testament to the power of algebraic manipulation in solving mathematical equations. Understanding these principles is crucial for success in mathematics and various related fields. Therefore, option C, $x = \frac{4}{3}$, is the correct answer to the given equation.

Importance of Checking Your Answer

While we have confidently arrived at the solution $x = \frac{4}{3}$ for the equation $-x + \frac{3}{7} = 2x - \frac{25}{7}$, it is always a prudent practice to verify our answer. Checking our solution helps ensure that we haven't made any errors in our calculations and that the value we found for x indeed satisfies the original equation. To check our answer, we substitute $x = \frac{4}{3}$ back into the original equation and see if both sides of the equation are equal. Let's perform this substitution:

$-(\frac{4}{3}) + \frac{3}{7} = 2(\frac{4}{3}) - \frac{25}{7}$

First, let's simplify the left-hand side (LHS):

$-\frac{4}{3} + \frac{3}{7}$

To add these fractions, we need a common denominator, which is 21. So, we convert the fractions:

$-\frac{4}{3} * \frac{7}{7} + \frac{3}{7} * \frac{3}{3} = -\frac{28}{21} + \frac{9}{21}$

Now, we can add the numerators:

$\frac{-28 + 9}{21} = -\frac{19}{21}$

So, the LHS simplifies to $-\frac{19}{21}$.

Next, let's simplify the right-hand side (RHS):

$2(\frac{4}{3}) - \frac{25}{7} = \frac{8}{3} - \frac{25}{7}$

Again, we need a common denominator, which is 21. Converting the fractions:

$\frac{8}{3} * \frac{7}{7} - \frac{25}{7} * \frac{3}{3} = \frac{56}{21} - \frac{75}{21}$

Subtracting the numerators:

$\frac{56 - 75}{21} = -\frac{19}{21}$

So, the RHS also simplifies to $-\frac{19}{21}$.

Since the LHS and RHS both equal $-\frac{19}{21}$ when we substitute $x = \frac{4}{3}$, we have verified that our solution is correct. This process of checking the answer is a valuable step in problem-solving, as it provides assurance that the solution is accurate and that no mistakes were made during the calculation process.

Conclusion: Mastering Linear Equations

In this comprehensive guide, we have explored the step-by-step process of solving the linear equation $-x + \frac{3}{7} = 2x - \frac{25}{7}$. We began by understanding the structure of the equation, identifying it as a linear equation in one variable, x. We then embarked on a journey to isolate x, employing fundamental algebraic techniques. The first step involved combining like terms, where we moved all the x terms to one side of the equation. This was achieved by adding x to both sides, resulting in the equation $\frac{3}{7} = 3x - \frac{25}{7}$. Next, we focused on isolating the x term further by moving the constant term to the other side. We added $\frac{25}{7}$ to both sides, simplifying the equation to $4 = 3x$. Finally, we solved for x by dividing both sides by 3, which yielded the solution $x = \frac{4}{3}$. This solution corresponds to option C in the original question. To ensure the accuracy of our solution, we emphasized the importance of checking the answer. By substituting $x = \frac{4}{3}$ back into the original equation, we verified that both sides of the equation were equal, confirming the correctness of our solution. This process of checking is a crucial step in problem-solving, providing confidence in the result and identifying any potential errors. The techniques we have discussed, including combining like terms, moving constants, and dividing to isolate the variable, are essential skills for solving linear equations. Mastering these techniques not only enables you to solve specific equations but also lays a strong foundation for tackling more complex mathematical problems in the future. Linear equations are ubiquitous in various fields, making their understanding and solution a valuable asset. Therefore, the ability to confidently solve linear equations is a testament to your mathematical proficiency and problem-solving skills.