Solving 7x - 2(x - 10) = 40 A Step-by-Step Guide
Introduction
When faced with the equation 7x - 2(x - 10) = 40, finding the solution might seem daunting at first. However, by systematically applying the principles of algebra, we can unravel the equation and determine the value of x that satisfies it. This comprehensive guide will walk you through each step of the solution process, providing clear explanations and insights along the way. Whether you're a student honing your math skills or simply seeking to refresh your knowledge, this article will equip you with the tools to confidently tackle similar algebraic problems.
In this article, we will delve into the step-by-step process of solving the equation. We will begin by simplifying the equation through distribution and combining like terms. Then, we will isolate the variable x on one side of the equation, employing inverse operations to maintain balance. Finally, we will arrive at the solution for x and verify its correctness. By understanding the underlying principles and techniques, you will gain a solid foundation for solving a wide range of algebraic equations. So, let's embark on this mathematical journey together and unlock the solution to the equation 7x - 2(x - 10) = 40.
Step 1: Distribute the -2
The first crucial step in solving the equation 7x - 2(x - 10) = 40 involves simplifying the expression by distributing the -2 across the terms within the parentheses. This process eliminates the parentheses and sets the stage for combining like terms. Remember, the distributive property states that a(b + c) = ab + ac. Applying this principle to our equation, we multiply -2 by both x and -10:
7x - 2 * x - 2 * (-10) = 40
This yields:
7x - 2x + 20 = 40
By carefully distributing the -2, we have successfully removed the parentheses and transformed the equation into a more manageable form. This step is critical because it allows us to combine like terms in the subsequent step, bringing us closer to isolating the variable x. The ability to accurately distribute is a fundamental skill in algebra, and mastering it will significantly enhance your problem-solving abilities. Now that we have eliminated the parentheses, we can proceed to combine the x terms and further simplify the equation. This meticulous approach ensures that we maintain accuracy and clarity throughout the solution process.
Step 2: Combine Like Terms
After distributing the -2 in the equation 7x - 2x + 20 = 40, the next logical step is to combine like terms. In this case, the like terms are the terms containing the variable x, namely 7x and -2x. Combining these terms simplifies the equation and brings us closer to isolating x. To combine like terms, we simply add or subtract their coefficients. In this instance, we subtract 2 from 7:
(7 - 2)x + 20 = 40
This simplifies to:
5x + 20 = 40
By combining the like terms, we have reduced the equation to a more concise and manageable form. This step is crucial because it consolidates the x terms, making it easier to isolate x in the following steps. The ability to identify and combine like terms is a fundamental algebraic skill that streamlines the process of solving equations. Now that we have simplified the equation to 5x + 20 = 40, we can proceed to isolate x by performing inverse operations. This systematic approach ensures that we maintain the balance of the equation while working towards the solution.
Step 3: Isolate the Variable
Now that we have simplified the equation to 5x + 20 = 40, our goal is to isolate the variable x on one side of the equation. To achieve this, we need to undo the operations that are being performed on x. In this case, x is being multiplied by 5 and then 20 is being added. We will use inverse operations to reverse these actions. The first step is to subtract 20 from both sides of the equation. This maintains the balance of the equation and isolates the term with x:
5x + 20 - 20 = 40 - 20
This simplifies to:
5x = 20
Next, to completely isolate x, we need to undo the multiplication by 5. We do this by dividing both sides of the equation by 5:
5x / 5 = 20 / 5
This gives us:
x = 4
By performing these inverse operations, we have successfully isolated the variable x and determined its value. This process of isolating the variable is a core technique in algebra, and understanding it is essential for solving a wide variety of equations. We have now found a potential solution for x, but it's crucial to verify our answer to ensure its correctness. In the next step, we will substitute our solution back into the original equation to confirm that it satisfies the equation.
Step 4: Verify the Solution
After finding a potential solution for x, it's imperative to verify its correctness. This step ensures that the value we obtained indeed satisfies the original equation. To verify the solution x = 4, we substitute it back into the original equation 7x - 2(x - 10) = 40:
7 * 4 - 2(4 - 10) = 40
Now, we simplify the expression following the order of operations:
28 - 2(-6) = 40
28 + 12 = 40
40 = 40
As we can see, the left side of the equation equals the right side when x = 4. This confirms that our solution is correct. Verification is a crucial step in the problem-solving process, as it helps to identify any errors that may have occurred during the solution process. By substituting the solution back into the original equation, we can be confident that our answer is accurate. Now that we have verified our solution, we can confidently state that the value of x that satisfies the equation 7x - 2(x - 10) = 40 is indeed 4. This systematic approach to solving equations, which includes verification, ensures both accuracy and understanding.
Conclusion
In this comprehensive guide, we have successfully navigated the process of solving the equation 7x - 2(x - 10) = 40. By meticulously applying algebraic principles, we have determined that the solution is x = 4. Our journey began with distributing the -2 across the terms within the parentheses, followed by combining like terms to simplify the equation. We then isolated the variable x by employing inverse operations, ensuring the balance of the equation throughout the process. Finally, we verified our solution by substituting it back into the original equation, confirming its accuracy.
This step-by-step approach not only provides the solution but also reinforces the fundamental concepts of algebra. By understanding the logic behind each step, you can confidently tackle similar equations and enhance your problem-solving skills. Remember, the key to success in algebra lies in practice and a systematic approach. By mastering techniques such as distribution, combining like terms, and isolating variables, you can unlock the solutions to a wide range of mathematical challenges. Whether you're a student or simply someone looking to sharpen your math skills, the principles outlined in this guide will serve as a valuable resource. So, continue to explore the world of algebra, and embrace the power of mathematical reasoning.
The solution to the equation 7x - 2(x - 10) = 40 is D. x = 4.
