Solving 3(x-7)=42 A Step-by-Step Guide

Introduction: Understanding the Basics of Algebraic Equations

In the realm of mathematics, solving equations is a fundamental skill. Algebraic equations are mathematical statements that assert the equality of two expressions. These expressions often contain variables, which are symbols (usually letters) representing unknown values. Our goal in solving an equation is to find the value(s) of the variable(s) that make the equation true. This process involves manipulating the equation using algebraic principles until the variable is isolated on one side, revealing its solution. A simple example is the equation x + 2 = 5, where ‘x’ is the variable. To solve this, we subtract 2 from both sides, resulting in x = 3. This means that when x is 3, the equation holds true. Equations can range from simple linear forms to complex polynomials, and the techniques for solving them vary accordingly. Understanding the underlying principles of equation solving is crucial for success in higher-level math and various scientific disciplines. In this comprehensive guide, we will delve into the step-by-step process of solving the equation 3(x-7)=42, providing a clear and detailed explanation for each step involved. This particular equation is a linear equation, which means the variable 'x' is raised to the power of 1. Linear equations are among the simplest types of equations to solve, making them an excellent starting point for understanding algebraic manipulation. We will explore the properties of equality, the distributive property, and other key concepts that are essential for tackling this and similar equations. By the end of this guide, you will not only understand how to solve 3(x-7)=42 but also gain a broader understanding of the principles behind solving algebraic equations, empowering you to tackle more complex problems with confidence.

Step-by-Step Solution of 3(x-7)=42

In this section, we will embark on a step-by-step journey to solve the equation 3(x-7)=42. Each step will be meticulously explained, ensuring that you grasp the underlying logic and mathematical principles involved. We will begin by applying the distributive property to eliminate the parentheses, then isolate the variable term, and finally solve for 'x'. This structured approach will not only lead us to the solution but also enhance your understanding of algebraic problem-solving techniques. The first step in solving 3(x-7)=42 involves dealing with the parentheses. To do this, we employ the distributive property. The distributive property states that a(b + c) = ab + ac. In our equation, 3 is being multiplied by the expression (x - 7). Applying the distributive property, we multiply 3 by both 'x' and '-7'. This gives us 3 * x - 3 * 7, which simplifies to 3x - 21. So, the equation now becomes 3x - 21 = 42. This step is crucial because it transforms the equation into a more manageable form, removing the parentheses and allowing us to proceed with isolating the variable. The distributive property is a fundamental concept in algebra, and mastering its application is essential for solving a wide range of equations. By correctly applying this property, we have successfully expanded the left side of the equation, setting the stage for the next steps in our solution. The next key step in solving the equation 3x - 21 = 42 is to isolate the term containing the variable 'x'. This means we want to get the term '3x' by itself on one side of the equation. To achieve this, we need to eliminate the constant term, which is '-21' in this case. We can do this by using the addition property of equality. This property states that adding the same value to both sides of an equation maintains the equality. In our situation, we add 21 to both sides of the equation. This gives us 3x - 21 + 21 = 42 + 21. Simplifying both sides, we get 3x = 63. By adding 21 to both sides, we have successfully isolated the term '3x' on the left side of the equation. This step is vital because it brings us closer to solving for 'x'. Isolating the variable term is a common strategy in equation solving, and understanding how to do it efficiently is key to success in algebra. Now that we have 3x = 63, the final step is to solve for 'x' itself. The 'x' is currently being multiplied by 3. To isolate 'x', we need to perform the inverse operation, which is division. We use the division property of equality, which states that dividing both sides of an equation by the same non-zero value maintains the equality. In this case, we divide both sides of the equation by 3. This gives us (3x) / 3 = 63 / 3. Simplifying both sides, we find that x = 21. This is the solution to the equation 3(x-7)=42. It means that when x is 21, the equation holds true. We have successfully solved for 'x' by dividing both sides of the equation by its coefficient. This final step demonstrates the power of inverse operations in equation solving. By systematically applying the distributive property, the addition property of equality, and the division property of equality, we have arrived at the solution. In conclusion, the solution to the equation 3(x-7)=42 is x = 21. We arrived at this solution by meticulously following a step-by-step process that involved applying the distributive property to eliminate parentheses, adding 21 to both sides to isolate the variable term, and dividing both sides by 3 to solve for 'x'. This process exemplifies the fundamental principles of algebraic equation solving. To ensure our solution is correct, we can perform a verification step. This involves substituting the value we found for 'x' back into the original equation. If the equation holds true, then our solution is correct. Substituting x = 21 into the original equation 3(x-7)=42, we get 3(21-7)=42. Simplifying the expression inside the parentheses, we have 3(14)=42. Multiplying 3 by 14, we get 42 = 42. Since both sides of the equation are equal, our solution x = 21 is indeed correct. This verification step is a crucial part of the problem-solving process, as it confirms the accuracy of our work and provides confidence in our answer. By consistently verifying our solutions, we can avoid errors and strengthen our understanding of algebraic concepts. The ability to solve equations like 3(x-7)=42 is a fundamental skill in mathematics, with applications extending far beyond the classroom. Whether you're solving a physics problem, balancing a budget, or making calculations in everyday life, the principles of equation solving are invaluable. This step-by-step guide has provided you with the tools and understanding necessary to tackle similar equations with confidence. Remember to always follow a systematic approach, applying the properties of equality and inverse operations to isolate the variable and find the solution. With practice and persistence, you can master the art of equation solving and unlock the power of mathematics.

Common Mistakes to Avoid When Solving Equations

Solving equations is a fundamental skill in mathematics, but it's also an area where common mistakes can easily occur. Recognizing these pitfalls and understanding how to avoid them is crucial for achieving accurate solutions. In this section, we will explore some of the most frequent errors made when solving equations and provide strategies for preventing them. One of the most common mistakes in solving equations is an incorrect application of the distributive property. As we saw in solving 3(x-7)=42, the distributive property is essential for eliminating parentheses. However, it's easy to make errors if not applied carefully. For example, when distributing a negative number, students may forget to distribute the negative sign to all terms inside the parentheses. This can lead to a completely incorrect equation and, consequently, a wrong solution. To avoid this mistake, always double-check that you have multiplied the term outside the parentheses by every term inside, paying close attention to signs. Writing out each step explicitly can help minimize errors. Another frequent mistake arises from 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 performed on the other to preserve the equality. For instance, if you add a number to the left side, you must add the same number to the right side. Failing to do so disrupts the balance and leads to an incorrect solution. A common error in this category is adding or subtracting a term from only one side or dividing only one side by a number. To avoid this, always remember the golden rule of equation solving: what you do to one side, you must do to the other. This principle applies to all operations, including addition, subtraction, multiplication, and division. Keep your equation balanced, and you'll be on the right track. Errors in combining like terms are also a common source of mistakes. Like terms are terms that have the same variable raised to the same power. They can be combined by adding or subtracting their coefficients. For example, 3x + 2x are like terms and can be combined to 5x. However, 3x and 2x² are not like terms and cannot be combined. A frequent mistake is incorrectly combining terms that are not alike, leading to a simplified equation that is not equivalent to the original. To avoid this error, carefully identify like terms and combine only those terms. It can be helpful to rearrange the equation so that like terms are grouped together before combining them. This visual organization can reduce the chance of making a mistake. Another potential pitfall is making errors with signs, particularly when dealing with negative numbers. Sign errors can easily occur during addition, subtraction, multiplication, and division. A single sign error can throw off the entire solution. To minimize sign errors, it's essential to be meticulous and pay close attention to the rules of sign manipulation. For example, remember that subtracting a negative number is the same as adding the positive counterpart, and multiplying or dividing two negative numbers results in a positive number. Writing out each step carefully and double-checking your signs can help prevent these errors. Ignoring the order of operations is yet another mistake that can lead to incorrect solutions. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which operations should be performed. Failing to follow this order can result in incorrect simplification and an incorrect solution. For instance, in the expression 2 + 3 * 4, multiplication should be performed before addition. If addition is performed first, the result will be incorrect. To avoid this mistake, always adhere to the order of operations. If necessary, rewrite the equation to clarify the order in which operations should be performed. A final common mistake is not verifying the solution. As we demonstrated in solving 3(x-7)=42, verifying the solution is a crucial step in the problem-solving process. Substituting the solution back into the original equation ensures that it satisfies the equation. If the equation does not hold true, then an error has been made, and the solution is incorrect. To avoid this mistake, always take the time to verify your solution. This simple step can catch errors and provide confidence in your answer. In conclusion, solving equations requires careful attention to detail and a thorough understanding of algebraic principles. By being aware of common mistakes, such as incorrectly applying the distributive property, not performing operations on both sides, errors in combining like terms, sign errors, ignoring the order of operations, and not verifying the solution, you can significantly improve your accuracy and success in solving equations. Remember to always double-check your work, follow a systematic approach, and verify your answers. With practice and diligence, you can master the art of equation solving.

Conclusion: Mastering Equation Solving Skills

In conclusion, mastering the skills required to solve equations is a cornerstone of mathematical proficiency. Throughout this guide, we have delved into the step-by-step process of solving the equation 3(x-7)=42, explored common mistakes to avoid, and emphasized the importance of understanding the underlying algebraic principles. The ability to solve equations extends far beyond the classroom, finding applications in various fields, from science and engineering to finance and everyday problem-solving. Solving the equation 3(x-7)=42 provided a practical example of how to apply fundamental algebraic concepts. We began by using the distributive property to eliminate parentheses, transforming the equation into a more manageable form. We then employed the addition property of equality to isolate the variable term, followed by the division property of equality to solve for 'x'. The solution we arrived at, x = 21, was then verified by substituting it back into the original equation, ensuring its accuracy. This systematic approach highlights the importance of following a structured process when solving equations. Each step is built upon the previous one, and a clear understanding of the properties of equality is essential for maintaining balance and arriving at the correct solution. Moreover, we discussed several common mistakes that students often make when solving equations. These included errors in applying the distributive property, failing to perform operations on both sides, incorrectly combining like terms, sign errors, ignoring the order of operations, and neglecting to verify the solution. By being aware of these potential pitfalls, students can take proactive steps to avoid them, such as double-checking their work, writing out each step explicitly, and verifying their solutions. The importance of understanding the underlying algebraic principles cannot be overstated. Equation solving is not simply about memorizing steps; it's about comprehending the logic behind each operation. The distributive property, the properties of equality, and the order of operations are not just rules to be followed; they are fundamental concepts that govern algebraic manipulation. A deep understanding of these principles empowers students to tackle a wide range of equations with confidence and flexibility. Furthermore, practice plays a crucial role in mastering equation solving skills. The more equations you solve, the more comfortable you will become with the process. Practice helps solidify your understanding of the concepts and allows you to develop problem-solving strategies. It also helps you identify and correct any weaknesses in your understanding. Start with simple equations and gradually progress to more complex ones. Seek out opportunities to apply your equation solving skills in different contexts. The applications of equation solving are vast and varied. In science, equations are used to model physical phenomena and make predictions. In engineering, they are used to design structures and systems. In finance, they are used to analyze investments and manage risk. Even in everyday life, we use equation solving skills to balance budgets, calculate discounts, and make informed decisions. By mastering equation solving, you are not only developing a valuable mathematical skill but also enhancing your ability to think critically and solve problems in a variety of situations. In conclusion, mastering equation solving skills is a journey that requires understanding, practice, and perseverance. By following a systematic approach, avoiding common mistakes, and developing a deep understanding of algebraic principles, you can unlock the power of equations and apply them to solve a wide range of problems. Remember that mathematics is not just about finding the right answer; it's about developing a way of thinking. Equation solving is a process that cultivates logical reasoning, critical thinking, and problem-solving skills. Embrace the challenge, and you will reap the rewards of mathematical mastery.