Solving Linear Equations Step By Step Guide
In the realm of mathematics, solving equations is a fundamental skill. Linear equations, in particular, form the bedrock of numerous mathematical concepts and real-world applications. This article delves into the intricacies of solving linear equations, providing a step-by-step guide and illustrative examples to solidify your understanding. We will explore two specific equations, demonstrating the techniques involved in isolating the variable and arriving at the solution. Linear equations are algebraic expressions that involve a variable raised to the power of one. Solving linear equations involves finding the value of the variable that makes the equation true. This often entails isolating the variable on one side of the equation by performing the same operations on both sides. Understanding the properties of equality is crucial for solving linear equations effectively. The addition property of equality states that adding the same quantity to both sides of an equation does not change the solution. Similarly, the subtraction property of equality allows us to subtract the same quantity from both sides without altering the solution. These properties, along with the multiplication and division properties of equality, form the foundation for manipulating equations and isolating variables. In this comprehensive guide, we will not only dissect the methods for solving linear equations but also emphasize the importance of checking solutions to ensure accuracy. By mastering these techniques, you'll gain a solid foundation for tackling more complex mathematical problems. Let's embark on this journey of mathematical exploration and unlock the power of solving linear equations.
Equation 1 Solving -5x - 2 = -4x + 12
Let's tackle the first equation: -5x - 2 = -4x + 12. Our primary goal is to isolate the variable 'x' on one side of the equation. To achieve this, we'll employ a series of algebraic manipulations, ensuring that we maintain the equality throughout the process. The initial equation presents 'x' terms on both sides, complicating the isolation process. Our first step involves consolidating these terms to one side. We can accomplish this by adding 4x to both sides of the equation. This move strategically eliminates the '-4x' term on the right side while simultaneously combining the 'x' terms on the left side. The transformed equation now reads: -x - 2 = 12. Notice how adding 4x to both sides simplifies the equation by reducing the number of 'x' terms. Next, we shift our focus to isolating 'x' further. The constant term '-2' on the left side hinders our progress. To counteract this, we apply the addition property of equality once more, this time adding 2 to both sides. This action cancels out the '-2' on the left side, leaving us with '-x' as the sole term involving 'x'. The equation now stands as: -x = 14. We're tantalizingly close to our solution, but the negative sign attached to 'x' prevents us from declaring victory just yet. To resolve this, we multiply both sides of the equation by -1. This seemingly simple step flips the sign of both terms, effectively isolating 'x' and revealing its value. The final equation is: x = -14. And there we have it! We've successfully isolated 'x' and determined its value. However, our journey doesn't end here. A crucial step remains: verifying our solution.
To ensure the accuracy of our solution, we substitute x = -14 back into the original equation. This process, known as checking the solution, provides a safeguard against potential errors made during the algebraic manipulations. The original equation, -5x - 2 = -4x + 12, serves as our testing ground. We replace each instance of 'x' with '-14' and meticulously evaluate both sides of the equation. On the left-hand side, we have -5(-14) - 2. Following the order of operations, we first perform the multiplication: -5 multiplied by -14 equals 70. Then, we subtract 2, resulting in 68. So, the left-hand side evaluates to 68. Now, let's turn our attention to the right-hand side: -4(-14) + 12. Again, we begin with the multiplication: -4 multiplied by -14 equals 56. Adding 12 to this result gives us 68. Thus, the right-hand side also evaluates to 68. A moment of triumph! Both sides of the equation yield the same value, 68. This confirms that our solution, x = -14, is indeed correct. The satisfaction of verifying a solution cannot be overstated. It provides not only assurance but also a deeper understanding of the equation-solving process. In this case, we've successfully navigated the steps of isolating the variable and validating the result. This equation serves as a testament to the power of algebraic manipulation and the importance of meticulous verification. As we move on to the next equation, remember the principles we've applied here: consolidate terms, isolate the variable, and always check your solution.
Equation 2 Solving x / 6 + 2 = -5
Now, let's tackle the second equation: x / 6 + 2 = -5. This equation introduces a slightly different challenge, as the variable 'x' is divided by 6. However, the fundamental principles of solving linear equations remain the same: our goal is to isolate 'x' on one side of the equation. To begin, we address the constant term '+2' on the left side. To counteract its presence, we apply the subtraction property of equality, subtracting 2 from both sides of the equation. This action cancels out the '+2' on the left, leaving us with 'x / 6' as the sole term involving 'x'. The transformed equation now reads: x / 6 = -7. We've made significant progress in isolating 'x', but the division by 6 still needs to be addressed. To undo this division, we employ the multiplication property of equality. We multiply both sides of the equation by 6. This action effectively cancels out the division by 6 on the left side, leaving 'x' isolated. The equation now stands as: x = -42. With 'x' standing alone, we've successfully determined its value. However, as with the previous equation, we must subject our solution to the crucial test of verification.
To ensure the accuracy of our solution, we substitute x = -42 back into the original equation. This process of checking the solution serves as a vital safeguard against potential errors that may have crept in during the algebraic manipulations. The original equation, x / 6 + 2 = -5, serves as the arena for our verification. We replace 'x' with '-42' and meticulously evaluate both sides of the equation, adhering strictly to the order of operations. On the left-hand side, we have -42 / 6 + 2. First, we perform the division: -42 divided by 6 equals -7. Then, we add 2, resulting in -5. So, the left-hand side evaluates to -5. Now, let's turn our attention to the right-hand side, which is simply -5. A resounding confirmation! Both sides of the equation yield the same value, -5. This unequivocally confirms that our solution, x = -42, is indeed the correct value. The importance of this verification step cannot be overstated. It not only provides assurance that we've arrived at the correct solution but also reinforces our understanding of the equation-solving process. By substituting the solution back into the original equation, we're essentially retracing our steps, ensuring that each manipulation was valid and that no errors were introduced. In this case, we've successfully navigated the steps of isolating the variable and rigorously validating our result. This equation stands as a testament to the power of algebraic manipulation and the unwavering importance of meticulous verification. As we conclude our exploration of these two equations, remember the core principles we've applied: isolate the variable through strategic manipulations, and always, without exception, check your solution to ensure its accuracy.
Conclusion
In this comprehensive guide, we've explored the methods for solving linear equations, emphasizing the importance of isolating the variable and checking solutions. We've dissected two specific equations, demonstrating the techniques involved in algebraic manipulation and verification. Through these examples, we've highlighted the fundamental properties of equality and their crucial role in solving equations accurately. Solving linear equations is not merely a mathematical exercise; it's a foundational skill with far-reaching applications. From everyday problem-solving to advanced scientific calculations, the ability to manipulate equations and find solutions is invaluable. The techniques we've explored here form the building blocks for more complex mathematical concepts, such as systems of equations and inequalities. By mastering these fundamentals, you'll gain a solid foundation for future mathematical endeavors. The journey of solving equations is one of precision, logic, and perseverance. Each step demands careful consideration, and each solution warrants thorough verification. The satisfaction of arriving at the correct answer, coupled with the knowledge that your solution is accurate, is a reward in itself. As you continue your mathematical journey, remember the principles we've discussed: consolidate terms, isolate the variable, and always check your solution. With practice and dedication, you'll become proficient in solving linear equations and unlock the power of algebra. The world of mathematics awaits your exploration, and the ability to solve equations is your key to unlocking its mysteries. Embrace the challenge, persevere through the complexities, and revel in the elegance of mathematical solutions.