Solving The Equation: (3x + 4) / 2 = 9.5

Introduction

In this article, we will delve into the process of solving the algebraic equation (3x + 4) / 2 = 9.5. This type of equation is a fundamental concept in mathematics, and understanding how to solve it is crucial for more advanced mathematical topics. We will break down the steps in a clear and concise manner, ensuring that anyone, regardless of their mathematical background, can follow along. Our goal is to provide a comprehensive guide that not only solves the equation but also explains the underlying principles and reasoning. By the end of this discussion, you will be equipped with the knowledge and skills to tackle similar equations with confidence. This includes understanding the order of operations, the properties of equality, and how to isolate the variable to find its value. So, let's embark on this mathematical journey together and unravel the solution to the equation (3x + 4) / 2 = 9.5.

Understanding the Basics of Algebraic Equations

Before we dive into the specifics of solving our equation, it’s important to lay a foundation by understanding the basics of algebraic equations. At its core, an algebraic equation is a statement that asserts the equality of two expressions. These expressions can involve numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division. The goal when solving an equation is to find the value(s) of the variable(s) that make the equation true. In other words, we want to isolate the variable on one side of the equation to determine its numerical value. To achieve this, we employ various algebraic techniques that maintain the balance of the equation. Think of an equation as a balanced scale; whatever operation you perform on one side, you must also perform on the other to keep the scale balanced. This principle is the cornerstone of solving algebraic equations. The equation we are dealing with, (3x + 4) / 2 = 9.5, involves a single variable, ‘x,’ and our objective is to find the value of ‘x’ that satisfies this equation. This involves a series of steps, each carefully designed to simplify the equation while preserving its balance. These steps often include using the distributive property, combining like terms, and applying inverse operations. Understanding these basic concepts is essential for successfully solving not just this equation, but a wide range of algebraic problems. So, with this foundation in place, let's move on to the first step in solving our equation: eliminating the fraction.

Step 1: Eliminating the Fraction

The first step in solving the equation (3x + 4) / 2 = 9.5 is to eliminate the fraction. Fractions can make equations appear more complex, and removing them often simplifies the solving process. In this case, the entire expression (3x + 4) is being divided by 2. To counteract this division, we need to perform the inverse operation, which is multiplication. We will multiply both sides of the equation by 2. This ensures that the equation remains balanced, as we are performing the same operation on both sides. When we multiply the left side, (3x + 4) / 2, by 2, the division by 2 is canceled out, leaving us with just (3x + 4). On the right side, we multiply 9.5 by 2, which gives us 19. So, after this step, our equation transforms from (3x + 4) / 2 = 9.5 to 3x + 4 = 19. This new equation is significantly simpler to work with, as it no longer contains a fraction. This step highlights the importance of using inverse operations to isolate the variable. By multiplying both sides by 2, we effectively undid the division and brought us closer to isolating ‘x.’ This is a common strategy in solving algebraic equations and is applicable in many different scenarios. Now that we have eliminated the fraction, our next step will be to isolate the term containing ‘x’ by dealing with the constant term on the same side of the equation. This will further simplify the equation and bring us closer to our final solution.

Step 2: Isolating the Term with 'x'

Now that we have the equation 3x + 4 = 19, our next goal is to isolate the term containing ‘x,’ which in this case is 3x. To do this, we need to eliminate the constant term, which is +4, from the left side of the equation. Just as we used the inverse operation to eliminate the fraction, we will use another inverse operation here. The inverse of addition is subtraction, so we will subtract 4 from both sides of the equation. This maintains the balance of the equation, as we are performing the same operation on both sides. When we subtract 4 from the left side, the +4 and -4 cancel each other out, leaving us with just 3x. On the right side, we subtract 4 from 19, which gives us 15. So, after this step, our equation transforms from 3x + 4 = 19 to 3x = 15. This is a significant step forward, as we have successfully isolated the term containing ‘x’ on one side of the equation. This step demonstrates the importance of understanding inverse operations and how they can be used to simplify equations. By subtracting 4 from both sides, we effectively undid the addition and brought us closer to isolating ‘x.’ This is a crucial technique in algebra and is used extensively in solving various types of equations. With the term containing ‘x’ now isolated, our next step is to isolate ‘x’ itself. This will involve one final inverse operation, and it will lead us directly to the solution of the equation.

Step 3: Solving for 'x'

We've reached the crucial step of solving for 'x' in the equation 3x = 15. Here, ‘x’ is being multiplied by 3. To isolate ‘x,’ we need to perform the inverse operation of multiplication, which is division. We will divide both sides of the equation by 3. This maintains the balance of the equation, as we are performing the same operation on both sides. When we divide the left side, 3x, by 3, the multiplication by 3 is canceled out, leaving us with just ‘x.’ On the right side, we divide 15 by 3, which gives us 5. So, after this step, our equation transforms from 3x = 15 to x = 5. This final step reveals the solution to our equation: x = 5. This means that the value of ‘x’ that satisfies the original equation (3x + 4) / 2 = 9.5 is 5. This step underscores the power of using inverse operations to isolate the variable and find its value. By dividing both sides by 3, we effectively undid the multiplication and determined the solution for ‘x.’ This is the ultimate goal when solving algebraic equations, and it allows us to find the specific value that makes the equation true. Now that we have found the solution, it’s always a good practice to check our answer to ensure its accuracy. This is what we will do in the next section.

Step 4: Verifying the Solution

After finding a solution to an equation, it’s always a good practice to verify that the solution is correct. This helps to ensure that no mistakes were made during the solving process and that the solution indeed satisfies the original equation. To verify our solution, x = 5, we will substitute this value back into the original equation, which was (3x + 4) / 2 = 9.5. By substituting x = 5, we get (3 * 5 + 4) / 2. Now, we need to simplify this expression following the order of operations (PEMDAS/BODMAS). First, we perform the multiplication: 3 * 5 = 15. Then, we perform the addition within the parentheses: 15 + 4 = 19. Now, our expression becomes 19 / 2. Finally, we perform the division: 19 / 2 = 9.5. So, when we substitute x = 5 into the original equation, we get 9.5, which is equal to the right side of the original equation. This confirms that our solution, x = 5, is indeed correct. This verification step is crucial in mathematics, as it provides confidence in the solution and helps to catch any errors that might have occurred. It reinforces the understanding of the equation and the steps taken to solve it. By verifying our solution, we complete the process of solving the equation and ensure that our answer is accurate. This practice is highly recommended for all algebraic problems, as it adds a layer of certainty to the solution.

Conclusion

In this article, we have meticulously walked through the process of solving the equation (3x + 4) / 2 = 9.5. We began by understanding the basics of algebraic equations, emphasizing the importance of maintaining balance through inverse operations. We then methodically eliminated the fraction, isolated the term with 'x,' and finally solved for 'x,' arriving at the solution x = 5. To ensure accuracy, we verified our solution by substituting it back into the original equation, confirming its validity. This step-by-step approach highlights the systematic nature of solving algebraic equations and underscores the significance of each step. By mastering these techniques, you can confidently tackle a wide range of similar problems. The ability to solve algebraic equations is a fundamental skill in mathematics and has applications in various fields, from science and engineering to finance and economics. Understanding these concepts not only helps in academic pursuits but also enhances problem-solving skills in real-life situations. We hope this comprehensive guide has provided you with a clear understanding of how to solve this type of equation and has empowered you to approach future mathematical challenges with greater confidence. For further exploration of algebraic equations and related topics, you might find resources on websites like Khan Academy's Algebra Section to be incredibly beneficial.