Solving The Equation 13 = 2x - 2 - 5x A Step-by-Step Guide

In the realm of mathematics, solving equations is a fundamental skill. It's the bedrock upon which more complex concepts are built. This guide delves into the process of solving a specific linear equation, $13 = 2x - 2 - 5x$, providing a clear, step-by-step approach that demystifies the process. Whether you're a student grappling with algebra or simply seeking to brush up on your math skills, this guide will equip you with the knowledge and confidence to tackle similar problems. Linear equations, characterized by a variable raised to the power of one, are ubiquitous in mathematics and its applications. Mastering the art of solving them is crucial for success in higher-level mathematics, physics, engineering, and various other fields. This comprehensive guide aims to provide a solid foundation in solving linear equations, enabling you to approach them with ease and confidence. Let's embark on this mathematical journey together, unraveling the intricacies of solving equations. This journey will not only enhance your problem-solving skills but also deepen your understanding of mathematical principles. Remember, mathematics is not just about memorizing formulas; it's about understanding the underlying concepts and applying them effectively. As we delve deeper into the process of solving this equation, you'll gain a greater appreciation for the elegance and power of mathematics. So, let's begin our exploration of solving the equation $13 = 2x - 2 - 5x$. By the end of this guide, you'll be well-equipped to solve similar equations and tackle more challenging mathematical problems. The key is to approach each step methodically, paying close attention to the details and understanding the rationale behind each operation. With practice and perseverance, you'll become proficient in solving linear equations and confident in your mathematical abilities.

1. Simplify the Equation: Combining Like Terms

The first step in solving the equation is to simplify it by combining like terms. In the equation $13 = 2x - 2 - 5x$, we have two terms containing the variable 'x': $2x$ and $-5x$. These are like terms because they have the same variable raised to the same power (which is 1 in this case). We can combine these terms by adding their coefficients: $2 + (-5) = -3$. Therefore, $2x - 5x$ simplifies to $-3x$. Our equation now becomes $13 = -3x - 2$. This simplification step is crucial because it reduces the complexity of the equation, making it easier to solve. By combining like terms, we consolidate the variable terms into a single term, which streamlines the subsequent steps in the solving process. This principle of combining like terms applies to all algebraic equations, regardless of their complexity. It's a fundamental technique that simplifies the equation and paves the way for isolating the variable. In addition to combining the 'x' terms, we also need to address the constant terms. In this case, we have the constant term '-2' on the right side of the equation. We'll deal with this constant term in the next step when we isolate the variable. For now, we've successfully simplified the equation by combining the 'x' terms, resulting in a more manageable form: $13 = -3x - 2$. This step demonstrates the power of simplification in mathematics. By breaking down a complex equation into its constituent parts and combining like terms, we make it easier to understand and solve. This skill is essential for tackling more advanced mathematical problems in the future.

2. Isolate the Variable Term: Adding 2 to Both Sides

Our next goal is to isolate the term containing the variable 'x'. In the simplified equation $13 = -3x - 2$, the variable term is $-3x$. To isolate this term, we need to eliminate the constant term '-2' from the right side of the equation. We can achieve this by adding 2 to both sides of the equation. This is based on the fundamental principle of algebraic manipulation: whatever operation you perform on one side of the equation, you must also perform on the other side to maintain the equality. Adding 2 to both sides, we get: $13 + 2 = -3x - 2 + 2$. On the left side, $13 + 2$ simplifies to $15$. On the right side, $-2 + 2$ cancels out, leaving us with just $-3x$. Our equation now becomes $15 = -3x$. By adding 2 to both sides, we've successfully isolated the variable term $-3x$ on the right side of the equation. This step is a crucial milestone in the solving process. We're one step closer to finding the value of 'x'. The principle of adding the same quantity to both sides of an equation is a cornerstone of algebraic manipulation. It ensures that the equation remains balanced while allowing us to isolate the variable term. This technique is widely used in solving various types of equations, not just linear equations. By mastering this principle, you'll be well-equipped to tackle more complex algebraic problems. Remember, the key to solving equations is to systematically isolate the variable, step by step, using valid algebraic operations. In this case, adding 2 to both sides has brought us significantly closer to our goal.

3. Solve for x: Dividing Both Sides by -3

Now that we have isolated the variable term, $-3x$, our final step is to solve for x itself. The equation is currently $15 = -3x$. To isolate 'x', we need to undo the multiplication by -3. We can do this by dividing both sides of the equation by -3. Again, we apply the principle of performing the same operation on both sides to maintain the equality. Dividing both sides by -3, we get: $15 / (-3) = (-3x) / (-3)$. On the left side, $15 / (-3)$ simplifies to $-5$. On the right side, $(-3x) / (-3)$ simplifies to 'x', as the -3s cancel out. Our equation now becomes $-5 = x$. This is the solution to the equation! We have successfully isolated 'x' and found its value. The value of x is -5. This final step highlights the importance of inverse operations in solving equations. Multiplication and division are inverse operations, meaning they undo each other. By dividing both sides of the equation by -3, we effectively undid the multiplication by -3, isolating 'x'. This technique of using inverse operations is fundamental to solving all types of algebraic equations. Whether it's addition and subtraction or multiplication and division, understanding inverse operations is crucial for manipulating equations and isolating the variable. In this case, dividing by -3 was the key to unlocking the solution. We've now completed the process of solving the equation $13 = 2x - 2 - 5x$. By combining like terms, isolating the variable term, and dividing to solve for 'x', we've demonstrated a systematic approach to solving linear equations. This approach can be applied to a wide range of similar problems.

4. The Solution: x = -5

Therefore, the solution to the equation $13 = 2x - 2 - 5x$ is x = -5. We arrived at this solution by meticulously following a series of algebraic steps, each grounded in fundamental mathematical principles. First, we simplified the equation by combining like terms, consolidating the 'x' terms into a single term. This simplification made the equation more manageable and set the stage for the subsequent steps. Next, we isolated the variable term by adding 2 to both sides of the equation. This crucial step removed the constant term from the side containing the variable, bringing us closer to isolating 'x'. Finally, we solved for 'x' by dividing both sides of the equation by -3. This step undid the multiplication by -3, leaving us with the value of 'x'. The result, x = -5, is the solution to the equation. It's the value that, when substituted back into the original equation, makes the equation true. To verify our solution, we can substitute x = -5 back into the original equation: $13 = 2(-5) - 2 - 5(-5)$. Simplifying the right side, we get: $13 = -10 - 2 + 25$. Further simplification yields: $13 = 13$. Since the left side equals the right side, our solution x = -5 is indeed correct. This verification step is essential in problem-solving. It provides assurance that our solution is accurate and that we haven't made any errors in our calculations. By substituting the solution back into the original equation, we confirm that it satisfies the equation, thus validating our answer. The process of solving equations is not just about finding a number; it's about understanding the relationships between variables and constants and manipulating them using algebraic principles. Each step we take is guided by these principles, ensuring that we maintain the equality of the equation while moving closer to the solution. In this case, we've successfully navigated the steps and arrived at the correct solution, demonstrating the power of algebraic techniques in solving mathematical problems.

Conclusion: Mastering Equation Solving

In conclusion, mastering equation solving is a cornerstone of mathematical proficiency. The process we've outlined, from simplifying the equation to isolating the variable and solving for its value, provides a robust framework for tackling a wide range of linear equations. The equation $13 = 2x - 2 - 5x$ served as a valuable example, illustrating the key steps involved in solving for 'x'. By combining like terms, isolating the variable term, and applying inverse operations, we systematically arrived at the solution x = -5. This solution was then verified by substituting it back into the original equation, confirming its accuracy. The skills and techniques we've discussed extend far beyond this specific example. They are applicable to a vast array of mathematical problems, from basic algebra to more advanced calculus and differential equations. The ability to solve equations is not just a mathematical skill; it's a critical thinking skill that translates to various fields, including science, engineering, economics, and computer science. As you continue your mathematical journey, remember that practice is key. The more you solve equations, the more comfortable and confident you'll become with the process. Don't be afraid to make mistakes; they are valuable learning opportunities. Each error you encounter is a chance to deepen your understanding and refine your problem-solving skills. Embrace the challenge of solving equations, and you'll unlock a world of mathematical possibilities. The ability to manipulate equations, isolate variables, and find solutions is a powerful tool that will serve you well in your academic pursuits and beyond. So, continue to practice, explore, and expand your mathematical horizons. The world of mathematics is vast and fascinating, and mastering equation solving is a crucial step on the path to mathematical mastery.