Solving 2(5A + 2) - 2A = 3(2A + 3) + 7 A Step-by-Step Guide

Jul 11, 2025 by ADMIN 60 views

Solving the Equation 2(5A + 2) - 2A = 3(2A + 3) + 7: A Comprehensive Guide

Introduction

In this article, we will delve into the step-by-step solution of the algebraic equation 2(5A + 2) - 2A = 3(2A + 3) + 7. Algebraic equations form the bedrock of mathematics and are crucial in various fields, including physics, engineering, and computer science. Mastering the techniques to solve these equations is, therefore, essential for anyone pursuing studies or careers in these areas. Our discussion will break down each stage, providing a detailed explanation to ensure a solid grasp of the process. This guide aims to make solving such equations seem less daunting and more approachable, even for those who are relatively new to algebra. We will not only solve the equation but also emphasize the underlying principles that govern each step. Understanding these principles is vital for solving more complex problems in the future.

Expanding the Equation

The first crucial step in solving the equation 2(5A + 2) - 2A = 3(2A + 3) + 7 involves expanding both sides to remove the parentheses. This requires applying the distributive property, which states that a(b + c) = ab + ac. Applying this property to the left side of the equation, we multiply 2 by both terms inside the first parenthesis: 2 * 5A equals 10A, and 2 * 2 equals 4. Thus, 2(5A + 2) expands to 10A + 4. We then rewrite the left side of the equation, incorporating the -2A term, resulting in 10A + 4 - 2A. Moving to the right side of the equation, we again apply the distributive property, multiplying 3 by both terms inside the parenthesis: 3 * 2A equals 6A, and 3 * 3 equals 9. Consequently, 3(2A + 3) expands to 6A + 9. We rewrite the right side, including the +7 term, yielding 6A + 9 + 7. By expanding both sides, we transform the equation into a simpler form that is easier to manipulate and solve. This initial expansion is the foundation for the subsequent steps and is crucial for arriving at the correct solution.

Simplifying Both Sides

After expanding the equation 2(5A + 2) - 2A = 3(2A + 3) + 7, the next critical step is to simplify each side individually. This involves combining like terms, which are terms that contain the same variable raised to the same power. On the left side of the equation, we have the terms 10A and -2A, both of which contain the variable A raised to the first power. Combining these, we perform the operation 10A - 2A, which simplifies to 8A. The constant term on the left side is 4, which remains unchanged as there are no other constant terms to combine it with. Thus, the left side of the equation simplifies to 8A + 4. Now, let’s turn our attention to the right side of the equation. Here, we have the term 6A, which is the only term containing the variable A. The constant terms are 9 and 7. Combining these, we perform the operation 9 + 7, which equals 16. Therefore, the right side of the equation simplifies to 6A + 16. By simplifying both sides, we reduce the equation to a more manageable form: 8A + 4 = 6A + 16. This simplified form makes it easier to isolate the variable A and ultimately solve for its value. The process of combining like terms is a fundamental technique in algebra and is essential for efficiently solving equations.

Isolating the Variable

With the equation simplified to 8A + 4 = 6A + 16, the next key step is to isolate the variable A on one side of the equation. This is achieved by performing inverse operations to move terms across the equals sign. Our goal is to gather all terms containing A on one side and all constant terms on the other side. A common approach is to subtract the term with the smaller coefficient of A from both sides. In this case, we have 8A on the left and 6A on the right. Subtracting 6A from both sides maintains the equation’s balance while moving the A terms to the left. This operation gives us 8A - 6A + 4 = 6A - 6A + 16, which simplifies to 2A + 4 = 16. Now that we have all the A terms on the left side, we need to move the constant term from the left to the right. To do this, we subtract 4 from both sides of the equation. This maintains the balance and isolates the term with A. The equation becomes 2A + 4 - 4 = 16 - 4, which simplifies to 2A = 12. By performing these inverse operations, we have successfully isolated the variable term 2A on one side of the equation, setting the stage for the final step of solving for A. This process of isolating the variable is a cornerstone of algebraic manipulation.

Solving for A

Having successfully isolated the term 2A in the equation 2A = 12, the final step is to solve for the variable A. This involves performing one last inverse operation to get A by itself. In this case, A is being multiplied by 2. To undo this multiplication, we perform the inverse operation, which is division. We divide both sides of the equation by 2, ensuring that the equation remains balanced. This gives us 2A / 2 = 12 / 2. On the left side, 2A divided by 2 simplifies to A. On the right side, 12 divided by 2 equals 6. Therefore, the equation simplifies to A = 6. This is the solution to the original equation. It means that the value of A that satisfies the equation 2(5A + 2) - 2A = 3(2A + 3) + 7 is 6. We can verify this solution by substituting A = 6 back into the original equation and checking if both sides are equal. By performing this final step of division, we have successfully solved for A, demonstrating the power of inverse operations in isolating and determining the value of a variable in an algebraic equation.

Verification

To ensure the accuracy of our solution, it’s essential to verify it by substituting the value we found for A back into the original equation. This process, known as verification, provides a crucial check that helps to confirm the correctness of our algebraic manipulations. Our solution was A = 6. The original equation is 2(5A + 2) - 2A = 3(2A + 3) + 7. We will substitute A with 6 in this equation and simplify both sides separately.

Left Side Verification

Let's start by substituting A = 6 into the left side of the original equation, which is 2(5A + 2) - 2A. Replacing A with 6, we get 2(5 * 6 + 2) - 2 * 6. First, we simplify inside the parentheses: 5 * 6 equals 30, so we have 2(30 + 2) - 2 * 6. Next, we add 30 and 2 inside the parentheses, which gives us 2(32) - 2 * 6. Now, we perform the multiplications: 2 * 32 equals 64, and 2 * 6 equals 12. So, we have 64 - 12. Finally, subtracting 12 from 64, we get 52. Therefore, the left side of the equation, when A = 6, simplifies to 52. This value will be compared with the simplified value of the right side to check for consistency. The step-by-step simplification ensures that we follow the correct order of operations, leading to an accurate evaluation of the left side of the equation.

Right Side Verification

Now, we will substitute A = 6 into the right side of the original equation, which is 3(2A + 3) + 7. Replacing A with 6, we get 3(2 * 6 + 3) + 7. First, we simplify inside the parentheses: 2 * 6 equals 12, so we have 3(12 + 3) + 7. Next, we add 12 and 3 inside the parentheses, which gives us 3(15) + 7. Now, we perform the multiplication: 3 * 15 equals 45. So, we have 45 + 7. Finally, adding 45 and 7, we get 52. Therefore, the right side of the equation, when A = 6, also simplifies to 52. Comparing this result with the simplified value of the left side, we see that both sides are equal. This consistency confirms the correctness of our solution A = 6. The systematic approach to simplifying the right side, following the order of operations, ensures that the verification process is accurate and reliable.

Conclusion of Verification

Upon substituting A = 6 into both sides of the original equation, 2(5A + 2) - 2A = 3(2A + 3) + 7, we found that both sides simplify to 52. This equality confirms that our solution, A = 6, is indeed correct. The left side of the equation, 2(5A + 2) - 2A, evaluated to 52 when A was replaced with 6. Similarly, the right side of the equation, 3(2A + 3) + 7, also evaluated to 52 when A was replaced with 6. The fact that both sides of the equation yield the same value after substitution provides strong evidence that our algebraic manipulations were accurate and that we have successfully solved the equation. This verification step is a crucial part of the problem-solving process in algebra. It not only confirms the solution but also helps in identifying any potential errors made during the solving process. By consistently verifying our solutions, we can build confidence in our algebraic skills and ensure the accuracy of our results.

Conclusion

In conclusion, solving the equation 2(5A + 2) - 2A = 3(2A + 3) + 7 involves a series of systematic steps, each building upon the previous one. We began by expanding both sides of the equation using the distributive property, which allowed us to remove the parentheses and reveal the individual terms. Next, we simplified each side separately by combining like terms, reducing the equation to a more manageable form. This step is crucial for clarity and for making subsequent manipulations easier. We then proceeded to isolate the variable A on one side of the equation by using inverse operations, a fundamental technique in algebra. This involved moving terms across the equals sign while maintaining the balance of the equation. Finally, we solved for A by performing the necessary division to isolate the variable completely. Our solution was A = 6. To ensure the correctness of our solution, we performed a verification step, substituting A = 6 back into the original equation. We found that both sides of the equation simplified to the same value, 52, which confirmed the accuracy of our solution. This comprehensive approach to solving algebraic equations not only yields the correct answer but also reinforces the understanding of the underlying principles and techniques. Mastering these skills is essential for success in algebra and related fields.