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

Introduction: Mastering Algebraic Equations

In the realm of mathematics, solving equations is a fundamental skill, and this article delves into the step-by-step process of tackling the equation 10 - (x+3)/2 = 8. Algebraic equations like this one are the building blocks of more advanced mathematical concepts, and mastering their solution is crucial for anyone venturing into higher mathematics, physics, engineering, or even everyday problem-solving. This article aims to provide a comprehensive guide, breaking down each step with clarity and ensuring that even those new to algebra can follow along. We will explore the underlying principles, the order of operations, and the techniques required to isolate the variable and find the solution. By the end of this guide, you will not only understand how to solve this specific equation but also gain a broader understanding of algebraic problem-solving strategies that can be applied to a wide range of mathematical challenges. This is more than just finding an answer; it's about building a solid foundation in mathematical thinking and developing the confidence to approach complex problems with a structured and methodical approach. So, let's embark on this journey of mathematical exploration and unravel the mystery behind solving this equation.

Understanding the Basics of Algebraic Equations

Before we dive into solving the equation, it's crucial to understand the basic principles of algebraic equations. An algebraic equation is a mathematical statement that asserts the equality of two expressions. These expressions contain variables, which are symbols (usually letters like 'x', 'y', or 'z') that represent unknown quantities. The goal of solving an equation is to find the value(s) of the variable(s) that make the equation true. In our case, the equation 10 - (x+3)/2 = 8 involves the variable 'x', and our task is to determine the value of 'x' that satisfies this equation. To achieve this, we employ various algebraic techniques, such as addition, subtraction, multiplication, and division, while maintaining the equality of both sides of the equation. The key principle here is that whatever operation we perform on one side of the equation, we must also perform on the other side to preserve the balance. This ensures that the solution we find is indeed a valid solution. Understanding this fundamental principle is crucial for successfully navigating the world of algebraic equations. This foundational knowledge empowers us to manipulate equations strategically and arrive at the correct solution with confidence.

Step-by-Step Solution: Unraveling the Equation

Step 1: Isolate the Term with the Variable

The first step in solving the equation 10 - (x+3)/2 = 8 is to isolate the term that contains the variable 'x'. This means we want to get the term (x+3)/2 by itself on one side of the equation. To do this, we need to eliminate the constant term 10 from the left side. We can achieve this by subtracting 10 from both sides of the equation. Remember, whatever operation we perform on one side, we must do the same on the other side to maintain the balance. This is a fundamental principle of equation solving. Subtracting 10 from both sides gives us: 10 - (x+3)/2 - 10 = 8 - 10. Simplifying this, we get -(x+3)/2 = -2. Now, we have successfully isolated the term containing 'x', which is a crucial step towards finding the solution. This step demonstrates the importance of strategic manipulation in algebra, where we aim to simplify the equation step by step until we can easily determine the value of the variable.

Step 2: Eliminate the Fraction

Now that we have -(x+3)/2 = -2, the next step is to eliminate the fraction. Fractions can often complicate the equation-solving process, so getting rid of them is usually a good move. In this case, the term (x+3) is being divided by 2. To eliminate this division, we can multiply both sides of the equation by 2. This is another application of the principle that whatever operation we perform on one side, we must also perform on the other side. Multiplying both sides by 2 gives us: 2 * [-(x+3)/2] = 2 * (-2). Simplifying this, the 2 in the numerator and the 2 in the denominator on the left side cancel out, leaving us with -(x+3) = -4. This step highlights how strategic multiplication can simplify equations by removing fractions, making the subsequent steps easier to manage. By eliminating the fraction, we've brought the equation closer to a form where we can directly isolate the variable 'x'.

Step 3: Distribute the Negative Sign

We've arrived at the equation -(x+3) = -4. The next step involves distributing the negative sign on the left side of the equation. The negative sign in front of the parentheses effectively means we are multiplying the entire expression inside the parentheses by -1. This is a crucial step because it allows us to remove the parentheses and further simplify the equation. To distribute the negative sign, we multiply each term inside the parentheses by -1. So, -(x+3) becomes -x - 3. The equation now looks like this: -x - 3 = -4. This step demonstrates the importance of understanding the distributive property in algebra, which allows us to simplify expressions and equations by properly handling parentheses and signs. By distributing the negative sign, we've cleared another hurdle in our quest to isolate the variable 'x'. This manipulation brings us closer to a form where we can directly solve for 'x'.

Step 4: Isolate the Variable Term

After distributing the negative sign, our equation is -x - 3 = -4. The next crucial step is to isolate the variable term, which in this case is -x. To do this, we need to eliminate the constant term -3 from the left side of the equation. We can achieve this by adding 3 to both sides of the equation. This again reinforces the fundamental principle of maintaining balance by performing the same operation on both sides. Adding 3 to both sides gives us: -x - 3 + 3 = -4 + 3. Simplifying this, the -3 and +3 on the left side cancel each other out, leaving us with -x = -1. We have now successfully isolated the variable term -x. This step is a key milestone in the equation-solving process, as it brings us one step closer to directly determining the value of 'x'. Isolating the variable term is a strategic maneuver that simplifies the equation and prepares it for the final step of solving for the variable itself.

Step 5: Solve for the Variable

We've reached the penultimate step in our journey to solve the equation, and we're currently at -x = -1. To solve for the variable 'x', we need to get 'x' by itself without any negative sign. In this case, we have -x, which is the same as -1 * x. To eliminate the -1, we can divide both sides of the equation by -1. This maintains the balance of the equation while isolating 'x'. Dividing both sides by -1 gives us: (-x) / -1 = (-1) / -1. Simplifying this, a negative divided by a negative results in a positive, so we get x = 1. Therefore, the solution to the equation 10 - (x+3)/2 = 8 is x = 1. This final step demonstrates the power of algebraic manipulation in isolating the variable and arriving at the solution. We have successfully navigated through the equation, applying various techniques to unveil the value of 'x'.

Verification: Ensuring the Solution's Accuracy

Substituting the Solution Back into the Original Equation

After solving an equation, it's crucial to verify the solution to ensure its accuracy. This process involves substituting the value we found for the variable back into the original equation and checking if both sides of the equation are equal. In our case, we found that x = 1 is the solution to the equation 10 - (x+3)/2 = 8. To verify this, we will substitute x = 1 into the original equation. This gives us: 10 - (1+3)/2 = 8. Now, we need to simplify the left side of the equation to see if it equals 8. Following the order of operations (PEMDAS/BODMAS), we first 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 looks like this: 10 - 2 = 8. Finally, we perform the subtraction: 10 - 2 = 8. So, the left side of the equation simplifies to 8, which is equal to the right side of the equation. This confirms that our solution, x = 1, is indeed correct. Verifying the solution is a vital step in the equation-solving process, as it helps us catch any potential errors and ensures that we have arrived at the correct answer. This step reinforces the importance of accuracy and attention to detail in mathematical problem-solving.

Conclusion: Mastering Equation-Solving Skills

In conclusion, we have successfully navigated the process of solving the equation 10 - (x+3)/2 = 8. Through a series of logical steps, we isolated the variable 'x' and determined its value to be 1. We began by isolating the term containing the variable, then eliminated the fraction, distributed the negative sign, isolated the variable term, and finally, solved for the variable itself. Each step was carefully explained, emphasizing the underlying principles of algebraic manipulation and the importance of maintaining balance in the equation. Furthermore, we verified our solution by substituting it back into the original equation, confirming its accuracy. This comprehensive guide has not only provided a solution to a specific equation but has also highlighted the general strategies and techniques involved in solving algebraic equations. Mastering these skills is essential for success in mathematics and related fields. The ability to solve equations empowers us to tackle a wide range of problems, from simple to complex, and fosters a deeper understanding of mathematical concepts. With practice and perseverance, anyone can develop proficiency in equation-solving and unlock the power of algebra. This journey through the equation-solving process has been a testament to the beauty and logic of mathematics.