Solving 2(2x - 5) = 6x + 4 A Step By Step Guide With Verification

by ADMIN 66 views

Understanding the Problem

In this comprehensive guide, we will delve into the process of solving the equation 2(2x - 5) = 6x + 4. This is a fundamental algebraic problem that requires a systematic approach to find the value of the variable 'x' that satisfies the equation. Solving equations like this is a crucial skill in mathematics, applicable in various fields ranging from basic arithmetic to advanced calculus and real-world problem-solving scenarios. The steps involved include distributing, combining like terms, isolating the variable, and finally, solving for 'x'. Once we find a potential solution, we will verify our answer by substituting it back into the original equation to ensure it holds true. This verification step is essential to confirm the accuracy of our solution and to catch any potential errors made during the solving process. It reinforces the understanding of equation solving and enhances problem-solving skills in algebra.

Before we begin, let's break down the key components of this problem. We have a linear equation, which means the highest power of 'x' is 1. The equation involves parentheses, which indicate multiplication, and terms on both sides of the equals sign. Our goal is to isolate 'x' on one side of the equation, which means we need to undo any operations that are affecting 'x'. This involves using inverse operations, such as addition to undo subtraction and division to undo multiplication. The process requires careful attention to the order of operations and the properties of equality, which allow us to perform the same operation on both sides of the equation without changing its balance. By following a step-by-step approach, we can systematically solve the equation and arrive at the correct solution.

Algebraic equations form the backbone of many mathematical concepts and are integral to various applications in science, engineering, economics, and computer science. The ability to efficiently solve equations is not only important for academic success but also for practical problem-solving in real-life situations. From simple tasks like calculating expenses to complex simulations in scientific research, the principles of equation solving are universally applicable. By mastering these techniques, individuals can develop critical thinking and analytical skills that are valuable in a wide range of contexts. Moreover, the process of verifying solutions reinforces the importance of accuracy and attention to detail, which are essential attributes in any field. Therefore, a thorough understanding of how to solve equations, such as the one presented here, is a fundamental building block for further mathematical studies and practical applications.

Step-by-Step Solution

To solve the equation 2(2x - 5) = 6x + 4, we will follow these steps:

  1. Distribute the 2: The first step involves distributing the 2 on the left side of the equation. This means multiplying 2 by each term inside the parentheses: 2 * 2x and 2 * -5. This step is crucial because it removes the parentheses, allowing us to combine like terms in the subsequent steps. The distributive property is a fundamental concept in algebra, and its correct application is essential for simplifying equations. Failing to distribute correctly can lead to errors that propagate through the rest of the solution. Therefore, it is important to pay close attention to this step and ensure that each term inside the parentheses is multiplied by the constant outside.

    • 2(2x - 5) becomes 4x - 10.

  2. Rewrite the equation: After distributing, the equation now looks simpler. We have removed the parentheses and can proceed with the next steps to isolate the variable. Rewriting the equation in this simplified form makes it easier to see the terms and the operations involved. This step is a good practice to ensure clarity and avoid any confusion as we move forward. It also allows us to visually inspect the equation for any potential errors before proceeding further. By rewriting the equation, we set the stage for the next steps, which involve combining like terms and isolating the variable.

    • The equation becomes 4x - 10 = 6x + 4.

  3. Move the x terms to one side: To isolate 'x', we need to move all terms containing 'x' to one side of the equation. A common strategy is to move the term with the smaller coefficient of 'x' to the side with the larger coefficient. In this case, we can subtract 4x from both sides of the equation. This will eliminate the 'x' term on the left side and consolidate it on the right side. Remember, when we perform an operation on one side of the equation, we must perform the same operation on the other side to maintain the balance. This is a fundamental principle of equation solving, and it ensures that the equation remains valid throughout the process.

    • Subtract 4x from both sides: 4x - 10 - 4x = 6x + 4 - 4x.
    • This simplifies to -10 = 2x + 4.

  4. Move the constant terms to the other side: Now that we have the 'x' terms on one side, we need to move the constant terms to the other side. In this case, we will subtract 4 from both sides of the equation. This will isolate the term with 'x' on the right side. Just as in the previous step, maintaining the balance of the equation is crucial. We must perform the same operation on both sides to ensure that the equation remains valid. This step brings us closer to isolating 'x' and finding its value.

    • Subtract 4 from both sides: -10 - 4 = 2x + 4 - 4.
    • This simplifies to -14 = 2x.

  5. Solve for x: The final step is to solve for x. We have the equation -14 = 2x, which means 2 times 'x' equals -14. To find the value of 'x', we need to undo the multiplication by dividing both sides of the equation by 2. This will isolate 'x' and give us its value. Remember, division is the inverse operation of multiplication, and using it allows us to isolate the variable. This step completes the process of solving the equation and provides us with the solution.

    • Divide both sides by 2: -14 / 2 = 2x / 2.
    • This gives us x = -7.

Verification of the Solution

To verify the solution, we substitute x = -7 back into the original equation: 2(2x - 5) = 6x + 4. Verification is a critical step in the problem-solving process because it confirms the accuracy of our solution and helps us identify any errors that may have occurred during the solving process. By substituting the value of 'x' back into the original equation, we can check if both sides of the equation are equal. If they are, then our solution is correct. If they are not, then we know that there is an error somewhere in our calculations, and we need to go back and review each step to find the mistake. This process not only ensures the correctness of the answer but also reinforces our understanding of the equation and the steps involved in solving it.

  1. Substitute x = -7: Replace every instance of 'x' in the original equation with -7.

    • 2(2(-7) - 5) = 6(-7) + 4

  2. Simplify the left side: Start by simplifying the expression inside the parentheses, following the order of operations (PEMDAS/BODMAS). This involves performing the multiplication first and then the subtraction. It is important to pay attention to the signs of the numbers to avoid errors. Each step in the simplification process should be performed carefully to ensure accuracy. Once the parentheses are simplified, we can then multiply by the constant outside the parentheses.

    • 2(-14 - 5) = 6(-7) + 4
    • 2(-19) = 6(-7) + 4
    • -38 = 6(-7) + 4

  3. Simplify the right side: Now, simplify the right side of the equation. This involves multiplying 6 by -7 and then adding 4. Just like on the left side, it is important to follow the order of operations and pay attention to the signs. Simplifying each side of the equation separately helps to avoid confusion and makes it easier to compare the results. This step will give us a numerical value for the right side, which we can then compare to the simplified value of the left side.

    • -38 = -42 + 4
    • -38 = -38

  4. Compare both sides: If the left side equals the right side, the solution is correct. In this case, -38 = -38, which confirms that x = -7 is the correct solution. This final step is the moment of truth. If both sides of the equation are equal, it means that our solution satisfies the equation, and we can confidently conclude that we have solved the equation correctly. If the two sides are not equal, it indicates that we have made an error somewhere, and we need to revisit our steps to identify and correct the mistake. The verification process is an essential part of problem-solving, as it ensures the accuracy of our results.

Conclusion

In conclusion, the solution to the equation 2(2x - 5) = 6x + 4 is x = -7. We verified this by substituting -7 back into the original equation and confirming that both sides are equal. Solving equations and verifying the solutions are essential skills in algebra, providing a foundation for more advanced mathematical concepts. This step-by-step process demonstrates the importance of careful execution and verification in mathematical problem-solving. The ability to solve equations accurately and efficiently is a valuable skill that is applicable in many areas of life, from academic studies to practical problem-solving in the real world. By mastering these techniques, individuals can develop confidence in their mathematical abilities and enhance their critical thinking skills.