Solving 3x-5=4x-3 A Step-by-Step Guide

Introduction to Solving Linear Equations

In the realm of mathematics, linear equations form the bedrock of numerous concepts and applications. They represent the simplest yet most fundamental type of equation, characterized by a variable raised to the first power. Mastering the art of solving linear equations is not just a mathematical exercise; it's a crucial skill that unlocks doors to more advanced mathematical topics and real-world problem-solving scenarios. At the heart of any linear equation is the quest to find the value of the unknown variable that makes the equation true. This process involves employing a series of algebraic manipulations to isolate the variable on one side of the equation, revealing its solution. The equation 3x - 5 = 4x - 3 serves as an excellent example to illustrate these fundamental principles. This equation, seemingly straightforward, embodies the core concepts of linear equations and provides a practical pathway to understanding the techniques required to solve them. To successfully navigate the world of linear equations, one must grasp the significance of maintaining equality throughout the solution process. Every operation performed on one side of the equation must be mirrored on the other side, ensuring that the balance is preserved and the solution remains accurate. This principle of equality is the cornerstone of algebraic manipulation and underpins the validity of the solution. As we embark on this step-by-step guide, we will unravel the intricacies of solving 3x - 5 = 4x - 3, highlighting the key steps and underlying principles that govern the process. By understanding the rationale behind each step, you'll gain a deeper appreciation for the elegance and precision of mathematics, and you'll be well-equipped to tackle a wide range of linear equations with confidence and proficiency.

Step 1: Grouping Like Terms

In the journey of solving the equation 3x - 5 = 4x - 3, our first crucial step is to consolidate the like terms. Like terms, in algebraic parlance, are terms that share the same variable raised to the same power. In our equation, the terms 3x and 4x are like terms, as they both contain the variable 'x' raised to the power of one. Similarly, the constants -5 and -3 are also like terms, as they are both numerical values without any variable attached. The strategic grouping of like terms is pivotal because it simplifies the equation, making it more manageable and paving the way for the isolation of the variable. To achieve this grouping, we employ the fundamental principle of algebraic manipulation: performing the same operation on both sides of the equation to maintain equality. Our goal here is to bring all the 'x' terms to one side of the equation and all the constant terms to the other side. Let's begin by focusing on the 'x' terms. We can choose to move either 3x to the right side or 4x to the left side. For this example, let's opt to move 3x to the right side. To do this, we subtract 3x from both sides of the equation. This yields: (3x - 5) - 3x = (4x - 3) - 3x. On the left side, 3x and -3x cancel each other out, leaving us with just -5. On the right side, 4x - 3x simplifies to x. Our equation now transforms to -5 = x - 3. We have successfully grouped the 'x' terms on the right side, but we still have a constant term (-3) on the same side as 'x'. To isolate 'x' completely, we need to move this constant term to the left side. We can accomplish this by adding 3 to both sides of the equation. This gives us: -5 + 3 = (x - 3) + 3. On the left side, -5 + 3 simplifies to -2. On the right side, -3 and +3 cancel each other out, leaving us with just x. Our equation now elegantly simplifies to -2 = x. With this step, we have effectively grouped the like terms and made significant progress towards isolating the variable 'x'. The equation is now in a much simpler form, setting the stage for the final step of determining the value of 'x'.

Step 2: Isolating the Variable

The second step is where we isolate the variable. After executing the grouping of like terms, our equation has been streamlined to -2 = x. At this juncture, the variable 'x' is almost entirely isolated. A closer inspection reveals that 'x' is already standing alone on the right-hand side of the equation. This is a testament to the effectiveness of our previous step, where we meticulously grouped the like terms, paving the way for this straightforward isolation. In this particular instance, the isolation process is remarkably simple. We observe that 'x' is equal to -2, which directly provides us with the solution. There are no further algebraic manipulations required; the variable is already by itself, and its value is clearly defined. In other linear equations, the isolation step might involve additional operations such as division or multiplication to remove coefficients or constants that are attached to the variable. However, in this case, the equation's inherent structure has rendered the isolation process exceptionally concise. The equation -2 = x is not only the solution but also a clear declaration of the variable's value. It signifies that the value of 'x' that satisfies the original equation 3x - 5 = 4x - 3 is indeed -2. This elegant simplicity underscores the importance of strategic algebraic manipulation in simplifying equations and facilitating their solutions. By diligently grouping like terms, we have arrived at a point where the variable's value is self-evident, highlighting the power of mathematical precision and the beauty of a well-structured solution. With the variable now isolated, we can confidently proceed to the final step, which involves verifying our solution to ensure its accuracy.

Step 3: Verifying the Solution

Having arrived at a potential solution, the crucial final step in solving any equation is to verify its accuracy. This process involves substituting the solution back into the original equation to ensure that it holds true. In our case, we've determined that x = -2. To verify this, we'll substitute -2 for 'x' in the original equation, 3x - 5 = 4x - 3. Let's perform the substitution: 3(-2) - 5 = 4(-2) - 3. Now, we simplify both sides of the equation independently, following the order of operations. On the left-hand side, 3 multiplied by -2 equals -6, so we have -6 - 5. This simplifies to -11. On the right-hand side, 4 multiplied by -2 equals -8, so we have -8 - 3. This also simplifies to -11. Now, we compare the simplified results from both sides of the equation. We have -11 = -11. This is a true statement, indicating that our solution, x = -2, is indeed correct. The verification step serves as a safeguard against potential errors made during the solution process. By substituting the solution back into the original equation, we ensure that the equation remains balanced and that the value we've obtained truly satisfies the given conditions. In this instance, the verification process has confirmed that our solution is accurate and that we have successfully solved the equation 3x - 5 = 4x - 3. This step not only provides confidence in our answer but also reinforces the fundamental principle of equality that underpins all algebraic manipulations. With the solution verified, we can confidently conclude that x = -2 is the correct answer to the equation 3x - 5 = 4x - 3. This step-by-step journey, from grouping like terms to isolating the variable and finally verifying the solution, exemplifies the systematic approach required to solve linear equations effectively and accurately.

Conclusion: Solution and Implications

In conclusion, through a series of meticulous algebraic steps, we have successfully solved the linear equation 3x - 5 = 4x - 3. Our journey began with the strategic grouping of like terms, followed by the isolation of the variable, and culminated in the crucial verification of the solution. The final solution we arrived at is x = -2. This value represents the precise point at which the equation holds true, signifying that when -2 is substituted for 'x', both sides of the equation are equal. The implications of solving linear equations extend far beyond the realm of mathematical exercises. Linear equations serve as fundamental building blocks for more complex mathematical concepts and find widespread applications in various fields, including physics, engineering, economics, and computer science. They are used to model real-world scenarios, solve problems, and make predictions. The ability to confidently solve linear equations is therefore an essential skill for anyone pursuing studies or careers in these disciplines. The process we've undertaken in solving 3x - 5 = 4x - 3 exemplifies the systematic approach required to tackle linear equations effectively. By understanding the underlying principles of algebraic manipulation, such as maintaining equality and grouping like terms, we can simplify equations and isolate variables, ultimately revealing their solutions. Moreover, the verification step underscores the importance of accuracy and attention to detail in mathematical problem-solving. By substituting the solution back into the original equation, we can ensure that our answer is correct and that we have not made any errors along the way. This comprehensive approach not only yields the correct solution but also fosters a deeper understanding of the mathematical concepts involved. As we conclude this step-by-step guide, it's important to recognize that solving linear equations is not just about finding the right answer; it's about developing a logical and systematic approach to problem-solving. The skills and techniques we've explored here will serve as a solid foundation for tackling more challenging mathematical problems and for applying mathematical principles to real-world situations. With practice and persistence, anyone can master the art of solving linear equations and unlock the power of mathematics to solve problems and understand the world around us.