Solving And Verifying Linear Equations A Step-by-Step Guide
In the realm of mathematics, linear equations serve as fundamental building blocks for more complex concepts. Mastering the art of solving and verifying these equations is not just a crucial skill for academic success but also a valuable asset for problem-solving in various real-world scenarios. This comprehensive guide delves into the intricacies of linear equations, providing a step-by-step approach to solving them and verifying the solutions. Let's explore the equation $3(x-3)+13=2(x+5)$ as a practical example. Understanding how to solve linear equations is a crucial skill in mathematics, serving as the foundation for more advanced algebraic concepts. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. These equations are fundamental in various fields, including science, engineering, and economics, where they are used to model relationships between different quantities. The ability to solve linear equations accurately and efficiently is essential for problem-solving and decision-making in these disciplines. Before diving into the specifics of solving the equation $3(x-3)+13=2(x+5)$, it’s important to grasp the basic principles of linear equations. A linear equation can be represented in the form $ax + b = c$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. The goal is to isolate the variable on one side of the equation to determine its value. This process typically involves using algebraic operations to simplify the equation and maintain the equality. To effectively solve linear equations, one must be familiar with the properties of equality. These properties allow us to perform operations on both sides of the equation without changing its solution. The main properties include the addition property, the subtraction property, the multiplication property, and the division property. Each of these properties plays a crucial role in manipulating the equation to isolate the variable.
To solve the linear equation $3(x-3)+13=2(x+5)$, we embark on a systematic journey, guided by the principles of algebraic manipulation. Our primary objective is to isolate the variable $x$ on one side of the equation, thereby unveiling its value. This journey involves a series of steps, each carefully designed to bring us closer to our goal. Firstly, we confront the parentheses that encase the terms within the equation. To dismantle these barriers, we employ the distributive property, a fundamental principle that allows us to multiply a single term by each term within a set of parentheses. Applying this property to the left side of the equation, we multiply $3$ by both $x$ and $-3$, resulting in $3x - 9$. Similarly, on the right side, we multiply $2$ by both $x$ and $5$, yielding $2x + 10$. Our equation now stands as $3x - 9 + 13 = 2x + 10$, a simplified form that paves the way for further manipulation. With the parentheses dissolved, our focus shifts to combining like terms, a process that consolidates similar elements within the equation. On the left side, we identify $-9$ and $13$ as constants, terms that share the same variable-free nature. Combining these, we arrive at $4$, transforming the left side into $3x + 4$. Our equation now reads $3x + 4 = 2x + 10$, a more streamlined expression that brings us closer to isolating $x$. The next step in our quest is to gather all terms containing $x$ onto one side of the equation. To achieve this, we employ the principle of inverse operations, performing the opposite operation to eliminate terms from one side. In this case, we subtract $2x$ from both sides of the equation, effectively removing it from the right side and transferring its essence to the left. This subtraction yields $3x - 2x + 4 = 2x - 2x + 10$, which simplifies to $x + 4 = 10$. Our equation now boasts a single term containing $x$ on the left side, a significant milestone in our journey. Our penultimate step involves isolating $x$ by banishing the constant term that accompanies it. We once again invoke the principle of inverse operations, this time subtracting $4$ from both sides of the equation. This subtraction isolates $x$ on the left side and transforms the right side into a numerical value. The equation becomes $x + 4 - 4 = 10 - 4$, which simplifies to $x = 6$. Finally, we arrive at our destination, the solution to the equation. The value of $x$ that satisfies the equation $3(x-3)+13=2(x+5)$ is $6$. This value represents the point of equilibrium, the point where both sides of the equation hold equal weight. To summarize, solving the equation involves distributing constants across parentheses, combining like terms, moving variable terms to one side, and isolating the variable by performing inverse operations. Each step is critical in transforming the equation into a simplified form that reveals the value of $x$.
Having determined the solution to our linear equation, the crucial step of verification remains. Verification serves as the ultimate test, confirming the accuracy of our solution and ensuring that it harmonizes with the original equation. To verify the solution $x = 6$, we embark on a journey of substitution, replacing every instance of $x$ in the original equation with its numerical value. This substitution transforms the equation into a numerical statement, a statement whose truth or falsehood will validate our solution. Our original equation, $3(x-3)+13=2(x+5)$, now becomes $3(6-3)+13=2(6+5)$, a numerical expression poised for evaluation. We begin by simplifying the expressions within the parentheses, adhering to the order of operations, which prioritizes operations within parentheses. On the left side, $6 - 3$ simplifies to $3$, transforming the expression to $3(3) + 13$. Similarly, on the right side, $6 + 5$ simplifies to $11$, yielding $2(11)$. Our equation now reads $3(3) + 13 = 2(11)$, a step closer to revealing the truth. Next, we tackle the multiplication operations, multiplying the constants outside the parentheses by the terms within. On the left side, $3(3)$ evaluates to $9$, transforming the expression to $9 + 13$. On the right side, $2(11)$ evaluates to $22$, resulting in the equation $9 + 13 = 22$. With the multiplication complete, we move on to the addition operations, adding the remaining constants on each side. On the left side, $9 + 13$ sums to $22$, while the right side remains as $22$. Our equation now stands as $22 = 22$, a triumphant declaration of equality. The final step in our verification process is to analyze the resulting statement. The equation $22 = 22$ is a true statement, confirming that the left side is indeed equal to the right side. This equality signifies that our solution, $x = 6$, is the correct value that satisfies the original equation. The left-hand side (LHS) of the equation should equal the right-hand side (RHS) when the correct solution is substituted. If the LHS and RHS are equal, the solution is verified. In this case, substituting $x = 6$ into the original equation confirms the equality, thus validating the solution. In conclusion, the verification process serves as a vital safeguard, ensuring the accuracy of our solution and reinforcing our understanding of the equation. By substituting the solution back into the original equation and confirming the equality, we solidify our confidence in the correctness of our answer.
Based on our step-by-step solution and verification, we have confidently determined that the solution to the linear equation $3(x-3)+13=2(x+5)$ is $x = 6$. This solution represents the value that, when substituted into the equation, makes the equation true. Now, we turn our attention to selecting the correct choice from the options provided. The options typically present the solution in different formats, such as a set notation or a fill-in-the-blank format. Option A, "The solution set is $\square$," indicates that the solution should be expressed as a set. In this case, since we have a single solution, $x = 6$, the solution set would be ${6}$. We would fill in the box with the number $6$. This notation signifies that the set contains the single element $6$, which is the solution to the equation. Option B, "The solution is $\square$," suggests a fill-in-the-blank format where we directly input the value of the solution. In this case, we would simply fill in the box with the number $6$, representing the value of $x$ that satisfies the equation. The choice between Option A and Option B depends on the specific instructions and format provided in the problem. If the instructions explicitly state to provide the solution set, Option A is the appropriate choice. If the instructions simply ask for the solution, Option B is the more direct and concise option. In the context of our solution, both options are valid ways to represent the solution $x = 6$. The key is to understand the notation and format required by the problem and to accurately convey the solution in the specified manner. Ultimately, the goal is to communicate the solution clearly and effectively, ensuring that the answer is easily understood and correctly interpreted.
In this comprehensive guide, we have embarked on a journey to solve and verify the linear equation $3(x-3)+13=2(x+5)$. Through a step-by-step approach, we have unraveled the intricacies of the equation, employing the principles of algebraic manipulation to isolate the variable $x$. Our journey began with the distribution of constants across parentheses, followed by the combination of like terms, the strategic movement of variable terms to one side of the equation, and the final isolation of the variable through inverse operations. Each step was meticulously executed, guided by the fundamental rules of algebra. The solution we arrived at, $x = 6$, represented the value that, when substituted into the equation, would make the equation true. However, our journey did not end there. We recognized the paramount importance of verification, the process that serves as the ultimate test of our solution's accuracy. Through substitution, we replaced every instance of $x$ in the original equation with its numerical value, transforming the equation into a numerical statement. The subsequent simplification and evaluation of this statement led us to a resounding confirmation: the left side of the equation was indeed equal to the right side, validating our solution. This verification process not only solidified our confidence in the correctness of our answer but also reinforced our understanding of the equation's inherent properties. Finally, we turned our attention to the task of selecting the correct choice from the options provided. We explored the nuances of different formats, such as set notation and fill-in-the-blank, and emphasized the importance of adhering to the specific instructions and format stipulated in the problem. The ability to accurately convey the solution in the required format is a crucial skill in mathematics, ensuring that the answer is easily understood and correctly interpreted. In conclusion, mastering the art of solving and verifying linear equations is a cornerstone of mathematical proficiency. The ability to navigate the steps involved, from distribution and combination to isolation and verification, is essential for success in more advanced mathematical concepts and real-world problem-solving scenarios. This guide has provided a comprehensive framework for tackling linear equations, empowering you to approach these challenges with confidence and precision. Practice and persistence are key to honing these skills, transforming you into a proficient problem solver and a master of linear equations.
