Solving 10 - (x+3)/2 = 8 A Step-by-Step Guide

Introduction

In this article, we will provide a comprehensive, step-by-step solution to the equation 10 - (x+3)/2 = 8. This type of problem falls under the category of basic algebra, where we aim to find the value of the unknown variable, 'x', that makes the equation true. Understanding how to solve such equations is a fundamental skill in mathematics and is crucial for more advanced topics. Whether you're a student learning algebra for the first time or simply looking to refresh your skills, this guide will provide a clear and concise method for tackling similar problems. Let's dive in and explore the process of isolating 'x' and determining its value in this equation. By the end of this explanation, you will not only understand the solution to this specific problem but also gain valuable insights into the general techniques for solving linear equations.

Understanding the Equation

Before we jump into solving, let's break down the equation 10 - (x+3)/2 = 8 to ensure we understand each component. This equation is a linear equation because the highest power of the variable 'x' is 1. It involves several arithmetic operations: subtraction, addition, and division. The key to solving any equation is to isolate the variable, 'x', on one side of the equation. This means we need to undo the operations that are being applied to 'x'. In our equation, 'x' is part of the expression (x+3), which is then divided by 2, and finally, this result is subtracted from 10. To isolate 'x', we will reverse these operations in the correct order. This involves using the properties of equality, which state that we can perform the same operation on both sides of an equation without changing its validity. For instance, if we add or subtract the same number from both sides, or multiply or divide both sides by the same non-zero number, the equation remains balanced. Keep these principles in mind as we proceed through the steps to solve for 'x'. Understanding the structure of the equation is the first step towards successfully finding the solution.

Step 1: Isolate the Term with x

The primary goal in solving any algebraic equation is to isolate the variable on one side of the equation. In our case, the equation is 10 - (x+3)/2 = 8. The term containing 'x' is (x+3)/2. To isolate this term, we need to eliminate the '10' on the left side of the equation. We can achieve this by subtracting 10 from both sides of the equation. This maintains the balance of the equation, ensuring that both sides remain equal. Performing this operation gives us: 10 - (x+3)/2 - 10 = 8 - 10. Simplifying this, we get -(x+3)/2 = -2. This step is crucial because it brings us closer to isolating the variable 'x'. By removing the constant term on the left side, we are now left with an equation that only involves 'x' and the operations directly related to it. This simplification makes the subsequent steps easier to manage. The next step will focus on further isolating 'x' by dealing with the division by 2 and the negative sign.

Step 2: Eliminate the Fraction

After isolating the term with 'x', we have the equation -(x+3)/2 = -2. To further isolate 'x', we need to eliminate the fraction. The term (x+3) is being divided by 2, so to undo this division, we multiply both sides of the equation by 2. This is a fundamental algebraic principle: whatever operation you perform on one side of the equation, you must perform on the other side to maintain equality. Multiplying both sides by 2 gives us: 2 * [-(x+3)/2] = 2 * (-2). On the left side, the multiplication by 2 cancels out the division by 2, leaving us with -(x+3). On the right side, 2 multiplied by -2 equals -4. So, our equation now becomes -(x+3) = -4. This step is significant because it eliminates the fraction, making the equation simpler to work with. We are now closer to isolating 'x', and the next step will involve dealing with the negative sign and then isolating 'x' itself.

Step 3: Distribute the Negative Sign

Currently, our equation is -(x+3) = -4. The negative sign on the left side of the equation needs to be distributed to both terms inside the parentheses. This is equivalent to multiplying the entire expression (x+3) by -1. Distributing the negative sign, we get (-1) * x + (-1) * 3 = -4, which simplifies to -x - 3 = -4. This step is crucial because it removes the parentheses and allows us to further isolate 'x'. It's important to remember that distributing a negative sign changes the sign of each term inside the parentheses. Now, the equation is in a more manageable form, with 'x' only having a coefficient of -1 and a constant term of -3. The next step will involve moving the constant term to the right side of the equation, bringing us even closer to isolating 'x'.

Step 4: Isolate x by Adding 3 to Both Sides

Following the distribution of the negative sign, our equation stands as -x - 3 = -4. Our next objective is to isolate the term containing 'x', which in this case is -x. To achieve this, we need to eliminate the constant term, -3, on the left side of the equation. We can do this by adding 3 to both sides of the equation. This operation maintains the balance of the equation while moving us closer to our goal of isolating 'x'. Adding 3 to both sides gives us: -x - 3 + 3 = -4 + 3. Simplifying this, we find that the -3 and +3 on the left side cancel each other out, leaving us with -x. On the right side, -4 + 3 equals -1. Therefore, the equation now simplifies to -x = -1. This step is a significant advancement in our solution process, as we have successfully isolated the term with 'x' on one side of the equation. The next and final step will be to solve for 'x' by dealing with the negative sign.

Step 5: Solve for x

After isolating the term with 'x', we have the equation -x = -1. This equation tells us that the negative of 'x' is equal to -1. To find the value of 'x' itself, we need to eliminate the negative sign. We can do this by multiplying both sides of the equation by -1. This is based on the principle that multiplying a negative number by -1 results in its positive counterpart. Multiplying both sides of the equation -x = -1 by -1 gives us: (-1) * (-x) = (-1) * (-1). On the left side, (-1) * (-x) simplifies to x, and on the right side, (-1) * (-1) simplifies to 1. Therefore, the equation becomes x = 1. This final step provides us with the solution to the equation. We have successfully isolated 'x' and determined its value. In the context of the original problem, this means that substituting 1 for 'x' in the equation 10 - (x+3)/2 = 8 will make the equation true. We can verify this by plugging 1 back into the original equation and checking if both sides are equal.

Verification

To ensure our solution is correct, it's crucial to verify it by substituting the value we found for 'x' back into the original equation. Our original equation was 10 - (x+3)/2 = 8, and we found that x = 1. Substituting x = 1 into the equation, we get: 10 - (1+3)/2 = 8. Now, we simplify the equation step by step. First, we simplify the expression inside the parentheses: 1 + 3 = 4. So the equation becomes 10 - 4/2 = 8. Next, we perform the division: 4/2 = 2. The equation now is 10 - 2 = 8. Finally, we perform the subtraction: 10 - 2 = 8. This gives us 8 = 8, which is a true statement. Since both sides of the equation are equal when x = 1, our solution is correct. Verification is an essential step in problem-solving as it confirms the accuracy of our work and helps prevent errors. By substituting the solution back into the original equation, we can be confident that our answer is valid.

Conclusion

In conclusion, we have successfully solved the equation 10 - (x+3)/2 = 8 by following a step-by-step algebraic approach. We began by understanding the equation and identifying the operations affecting the variable 'x'. Then, we systematically isolated 'x' by reversing these operations, ensuring that we maintained the balance of the equation at each step. We first isolated the term containing 'x' by subtracting 10 from both sides, then eliminated the fraction by multiplying both sides by 2. After that, we distributed the negative sign and isolated 'x' further by adding 3 to both sides. Finally, we solved for 'x' by multiplying both sides by -1, which gave us the solution x = 1. To ensure the accuracy of our solution, we verified it by substituting x = 1 back into the original equation and confirming that both sides were equal. This process highlights the importance of understanding algebraic principles and applying them methodically to solve equations. The ability to solve such equations is a fundamental skill in mathematics and has wide-ranging applications in various fields. By mastering these techniques, you can confidently tackle more complex problems and deepen your understanding of mathematics.

Therefore, the correct answer is C. 1.