Solving For X Step By Step Guide

In the realm of mathematics, solving for x is a fundamental skill, forming the bedrock for more advanced concepts. This article delves into the process of solving a linear equation, providing a step-by-step guide and illuminating the underlying principles. Linear equations, characterized by a variable raised to the power of one, are ubiquitous in mathematics and its applications, making proficiency in solving them indispensable. Solving for x is a crucial skill in mathematics, and mastering it opens doors to a deeper understanding of algebraic concepts. In this comprehensive guide, we will explore a specific linear equation and demonstrate how to systematically solve for the unknown variable, x. The equation we'll be tackling is 5(x - 10) = 30 - 15x. This equation is a classic example of a linear equation, meaning that the highest power of x is 1. Understanding how to solve such equations is fundamental to algebra and beyond. We'll break down each step, ensuring clarity and comprehension, so you can confidently tackle similar problems. Before diving into the solution, it's essential to grasp the core principles that govern equation solving. The fundamental idea is to isolate the variable x on one side of the equation. We achieve this by performing operations on both sides of the equation, maintaining the equality. These operations include addition, subtraction, multiplication, and division. Each operation must be applied equally to both sides to preserve the balance of the equation. Think of it like a weighing scale; whatever you add or subtract from one side, you must do the same to the other to keep the scale balanced. This principle of maintaining equality is the cornerstone of equation solving. In the following sections, we'll apply this principle to the given equation, step by step, until we arrive at the solution for x. Remember, practice is key to mastering this skill. As you work through various examples, you'll develop an intuitive understanding of the process and become more proficient in solving for x. This skill is not only vital for mathematics but also for various other fields, including physics, engineering, and economics, where equations are used to model real-world phenomena. So, let's embark on this journey of solving for x and unlock the power of algebra!

To solve for x in the equation 5(x - 10) = 30 - 15x, we'll follow a series of algebraic steps, ensuring we maintain the balance of the equation at each stage. Our goal is to isolate x on one side of the equation. First, we'll tackle the parentheses by applying the distributive property. This involves multiplying the 5 by each term inside the parentheses: 5 * x and 5 * -10. This gives us 5x - 50. Now, our equation looks like this: 5x - 50 = 30 - 15x. Next, we want to gather all the x terms on one side of the equation. To do this, we'll add 15x to both sides. Adding 15x to the left side cancels out the -15x term on the right side, bringing all x terms to the left. This results in the equation: 5x + 15x - 50 = 30. Combining the x terms on the left side, 5x + 15x becomes 20x. So, the equation simplifies to 20x - 50 = 30. Now, we need to isolate the x term further. To do this, we'll add 50 to both sides of the equation. This cancels out the -50 on the left side, leaving us with 20x = 30 + 50. Simplifying the right side, 30 + 50 equals 80. Our equation now is 20x = 80. Finally, to solve for x, we'll divide both sides of the equation by 20. This isolates x on the left side: x = 80 / 20. Performing the division, 80 / 20 equals 4. Therefore, the solution to the equation 5(x - 10) = 30 - 15x is x = 4. This step-by-step process demonstrates how to systematically solve a linear equation. By applying the principles of algebraic manipulation, we successfully isolated x and found its value. Remember, the key is to maintain balance by performing the same operations on both sides of the equation. In the next section, we'll verify our solution to ensure its accuracy.

To ensure the accuracy of our solution, x = 4, it is crucial to verify the solution by substituting it back into the original equation. This process confirms that the value of x we found indeed satisfies the equation, meaning it makes the left side equal to the right side. Let's substitute x = 4 into the original equation: 5(x - 10) = 30 - 15x. Replacing x with 4, we get 5(4 - 10) = 30 - 15(4). Now, we'll simplify both sides of the equation separately. On the left side, we first evaluate the expression inside the parentheses: 4 - 10 = -6. So, the left side becomes 5(-6). Multiplying 5 by -6, we get -30. Therefore, the left side of the equation simplifies to -30. On the right side, we first perform the multiplication: 15(4) = 60. So, the right side becomes 30 - 60. Subtracting 60 from 30, we get -30. Therefore, the right side of the equation also simplifies to -30. Now, we compare the simplified left and right sides: -30 = -30. Since both sides are equal, our solution x = 4 is verified. This confirmation step is essential because it eliminates the possibility of errors made during the solving process. It provides confidence that the value we found for x is indeed the correct solution to the equation. In mathematical problem-solving, verification is a critical habit to develop. It not only ensures accuracy but also reinforces understanding of the concepts involved. By substituting the solution back into the original equation, we're essentially reversing the steps we took to solve it, which helps solidify our grasp of the algebraic principles. Furthermore, verification can help identify potential mistakes. If the left and right sides of the equation do not match after substitution, it indicates an error in the solving process, prompting us to re-examine our steps and correct the mistake. This iterative process of solving and verifying is a cornerstone of mathematical rigor. In conclusion, the verification step is not merely a formality; it's an integral part of the problem-solving process. It provides assurance, enhances understanding, and promotes accuracy. So, always remember to verify your solutions to ensure their correctness.

Understanding linear equations is of paramount importance in mathematics and various other disciplines. They form the foundation for more complex algebraic concepts and are widely used in real-world applications. A linear equation, as the name suggests, represents a straight line when graphed on a coordinate plane. This visual representation provides a powerful way to understand the relationship between variables. The simplicity of linear equations makes them amenable to analysis and interpretation. They are used to model a wide range of phenomena, from simple relationships like the cost of items based on quantity to more complex scenarios in physics, engineering, and economics. In mathematics, linear equations serve as building blocks for understanding systems of equations, inequalities, and linear functions. These concepts are essential for solving problems in calculus, linear algebra, and differential equations, which are core topics in higher mathematics. Moreover, the ability to solve linear equations is crucial for developing problem-solving skills. The systematic approach required to isolate the variable and find its value cultivates logical thinking and analytical abilities. These skills are transferable to other areas of mathematics and beyond. In practical applications, linear equations are used to model and solve problems in diverse fields. For instance, in physics, they describe the motion of objects under constant acceleration. In economics, they are used to analyze supply and demand relationships. In computer science, they are used in algorithms and data analysis. The versatility of linear equations stems from their ability to approximate many real-world relationships. While many phenomena are inherently non-linear, linear models can often provide a reasonable approximation over a limited range. This makes them a valuable tool for analysis and prediction. Furthermore, understanding linear equations fosters a deeper appreciation for the power of mathematical modeling. By translating real-world scenarios into mathematical equations, we can gain insights and make informed decisions. This process highlights the relevance of mathematics in everyday life and its ability to solve practical problems. In conclusion, mastering linear equations is not just an academic exercise; it's an essential skill that unlocks a world of mathematical and practical applications. From laying the groundwork for advanced mathematical concepts to modeling real-world phenomena, linear equations play a pivotal role in our understanding of the world.

When solving for x, particularly in linear equations, several common mistakes can trip up even seasoned mathematicians. Identifying these pitfalls and learning how to avoid them is crucial for ensuring accuracy and efficiency in problem-solving. One prevalent mistake is the incorrect application of the distributive property. This occurs when multiplying a number or variable by an expression enclosed in parentheses. The distributive property states that a(b + c) = ab + ac. A common error is forgetting to distribute the multiplication to all terms inside the parentheses. For example, in the equation 5(x - 10) = 30 - 15x, incorrectly distributing the 5 might lead to 5x - 10 instead of the correct 5x - 50. To avoid this mistake, always double-check that you've multiplied the term outside the parentheses by every term inside. Another frequent error is combining like terms incorrectly. Like terms are terms that have the same variable raised to the same power. For instance, 5x and 15x are like terms, but 5x and 5x² are not. When combining like terms, only add or subtract their coefficients (the numbers in front of the variables). A mistake here might involve adding 5x and 15 to get 20x, which is incorrect. To prevent this, carefully identify the like terms and ensure you're only combining them. A third common mistake is not performing the same operation on both sides of the equation. The fundamental principle of equation solving is maintaining balance. Any operation performed on one side must be mirrored on the other side to preserve the equality. For example, if you add 5 to the left side of the equation, you must also add 5 to the right side. Neglecting this principle leads to incorrect solutions. To avoid this, consciously remind yourself to perform the same operation on both sides at each step. Sign errors are also a common source of mistakes. A misplaced or dropped negative sign can drastically alter the solution. For example, when moving a term from one side of the equation to the other, remember to change its sign. To minimize sign errors, pay close attention to the signs of each term and double-check your work. Finally, failing to verify the solution is a significant oversight. As discussed earlier, verification is crucial for confirming the accuracy of your answer. Substituting the solution back into the original equation and checking if both sides are equal is a safeguard against errors. In conclusion, by being aware of these common mistakes and implementing strategies to avoid them, you can significantly improve your accuracy and efficiency in solving equations. Remember to distribute carefully, combine like terms correctly, maintain balance, watch out for sign errors, and always verify your solution.

To solidify your understanding of solving for x, it's essential to engage in practice problems. The more you practice, the more comfortable and confident you'll become in applying the concepts and techniques discussed. Here are a few practice problems for you to tackle: 1. 2(x + 3) = 10 2. 4x - 7 = 5 3. 3(x - 2) = 2x + 4 4. 6x + 10 = 2x - 6 5. -(x + 5) = 3x - 1 Let's break down a strategy for approaching these problems. First, carefully read the equation and identify the operations involved. Look for parentheses, like terms, and any potential simplifications. Second, apply the order of operations (PEMDAS/BODMAS) to guide your steps. This ensures you perform operations in the correct sequence: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Third, use algebraic manipulations to isolate the x term. This might involve distributing, combining like terms, adding or subtracting terms from both sides, and dividing or multiplying both sides by a constant. Fourth, remember to maintain the balance of the equation. Any operation performed on one side must be mirrored on the other side. Fifth, once you've found a solution for x, verify your answer by substituting it back into the original equation. If both sides are equal, your solution is correct. If not, re-examine your steps to identify any errors. As you work through these problems, pay attention to the common mistakes discussed earlier, such as incorrect distribution, combining unlike terms, and sign errors. Practice identifying and avoiding these pitfalls. Additionally, consider different approaches to solving the same problem. There's often more than one way to isolate the x term, and exploring alternative methods can deepen your understanding. For example, you might choose to distribute first or combine like terms first, depending on the equation's structure. If you encounter difficulties, don't hesitate to seek help from resources like textbooks, online tutorials, or teachers and classmates. Mathematics is a collaborative subject, and discussing problems with others can provide valuable insights. Remember, practice is the key to mastering any mathematical skill. The more you work through problems, the more natural and intuitive the process will become. So, grab a pencil and paper, and start solving! The solutions to these practice problems are provided below for you to check your work. Happy solving! Solutions: 1. x = 2 2. x = 3 3. x = 10 4. x = -4 5. x = -5

In conclusion, solving for x in the equation 5(x - 10) = 30 - 15x is a process that exemplifies the fundamental principles of algebra. Through a step-by-step approach, we systematically isolated x and arrived at the solution x = 4. We then rigorously verified this solution by substituting it back into the original equation, confirming its accuracy. This comprehensive guide has not only demonstrated the mechanics of solving a specific linear equation but has also underscored the broader importance of understanding linear equations in mathematics and its applications. We've highlighted the crucial role of verification in ensuring the correctness of solutions and the significance of avoiding common mistakes. Moreover, we've emphasized the need for consistent practice to solidify understanding and develop problem-solving skills. Linear equations serve as the cornerstone for more advanced mathematical concepts, and proficiency in solving them is essential for success in higher-level mathematics and related fields. The ability to manipulate equations, maintain balance, and isolate variables is a valuable skill that extends beyond the classroom. It's a skill that fosters logical thinking, analytical reasoning, and the ability to approach problems in a systematic and methodical manner. These are skills that are highly valued in various professions and aspects of life. Furthermore, the understanding of linear equations enables us to model and solve real-world problems. From calculating the cost of goods to analyzing scientific data, linear relationships are ubiquitous in the world around us. The ability to translate real-world scenarios into mathematical equations and solve them provides a powerful tool for understanding and predicting outcomes. As you continue your mathematical journey, remember that practice is the key to mastery. The more you engage with problems, the more intuitive the concepts will become. Don't be discouraged by challenges; instead, view them as opportunities for growth and learning. Seek help when needed, and collaborate with others to enhance your understanding. Mathematics is a journey of exploration and discovery, and the ability to solve equations is a fundamental step along that path. So, embrace the challenge, persevere through difficulties, and celebrate your successes. The world of mathematics awaits, and the skills you develop in solving equations will serve you well in your future endeavors. Congratulations on mastering this fundamental skill, and may your mathematical journey be filled with curiosity, discovery, and success!

Choose the correct answer

B. $x=4$