Solving Equations: Find 'y' When X = 12

Introduction

In this article, we'll break down how to evaluate the equation 2y = 10 - (1/2)x when x = 12. Equations might seem intimidating at first, but with a systematic approach, they become manageable. We'll walk through each step, ensuring you understand the underlying principles and can apply them to similar problems. Whether you're a student brushing up on algebra or simply curious about mathematical problem-solving, this guide is for you. This equation is a linear equation, a fundamental concept in algebra. Mastering the evaluation of such equations lays a strong foundation for more advanced mathematical topics. So, let's dive in and learn how to solve for 'y' when we know the value of 'x'. We will explore the importance of substitution, order of operations, and simplification to arrive at the correct solution. Remember, practice makes perfect, so feel free to try out this method with other equations as well. Our aim is to make the process clear and straightforward, so you can confidently tackle any equation that comes your way. So, let's begin our journey into the world of equation solving and discover the value of 'y'!

1. Understanding the Equation

Before we jump into solving, let's take a moment to understand the equation 2y = 10 - (1/2)x. This equation represents a linear relationship between two variables, 'x' and 'y'. The goal is to find the value of 'y' when 'x' is given as 12. Think of it like a puzzle – we have some information (the equation and the value of 'x'), and we need to find the missing piece (the value of 'y'). The equation tells us that twice 'y' is equal to 10 minus half of 'x'. This is crucial for setting up our solution. We need to follow the order of operations (PEMDAS/BODMAS) to ensure we solve the equation correctly. Understanding the structure of the equation allows us to approach the problem logically. We'll use substitution and simplification techniques to isolate 'y' on one side of the equation. This process involves replacing 'x' with its given value and then performing the necessary arithmetic operations. By grasping the relationship between 'x' and 'y' in the equation, we can visualize how changing 'x' will affect 'y'. This understanding is fundamental to solving not just this equation, but any linear equation. It's like learning the grammar of mathematics – once you understand the rules, you can communicate effectively with numbers and symbols. Now, let's move on to the next step, where we'll put our understanding into action and start solving the equation.

2. Substituting the Value of 'x'

The next step in evaluating the equation is to substitute the value of 'x', which is given as 12. This means we replace 'x' in the equation 2y = 10 - (1/2)x with 12. So, the equation becomes 2y = 10 - (1/2)(12). Substitution is a fundamental technique in algebra, allowing us to replace variables with their numerical values. This simplifies the equation and brings us closer to finding the value of 'y'. It's like filling in the blanks in a sentence – we're replacing a symbol with a concrete number. By substituting, we transform the equation from a general statement about the relationship between 'x' and 'y' to a specific problem that we can solve arithmetically. The key to successful substitution is accuracy. Make sure you replace 'x' with the correct value and maintain the integrity of the equation. This step is crucial because any mistake here will propagate through the rest of the solution. Think of it as the foundation of a building – if the foundation is weak, the entire structure is at risk. Once we've substituted correctly, we can move on to simplifying the equation using the order of operations. This involves performing the multiplication and subtraction in the correct sequence to arrive at a simplified form of the equation. So, with 'x' successfully replaced by 12, we're now ready to take the next step towards solving for 'y'.

3. Simplifying the Equation

Now that we've substituted x = 12 into the equation, we need to simplify the equation 2y = 10 - (1/2)(12). Following the order of operations (PEMDAS/BODMAS), we first perform the multiplication: (1/2)(12) = 6. So, the equation becomes 2y = 10 - 6. Simplification is the process of reducing an equation or expression to its simplest form. It involves performing arithmetic operations like multiplication, division, addition, and subtraction in the correct order. This step is essential for making the equation easier to solve. Think of it as decluttering – we're removing unnecessary complexity to reveal the underlying structure. The order of operations is crucial here. We must perform multiplication before subtraction to arrive at the correct answer. A common mistake is to subtract 10 and (1/2) first, which would lead to an incorrect result. Once we've completed the multiplication, the equation is much simpler: 2y = 10 - 6. Now, we can perform the subtraction: 10 - 6 = 4. This gives us the equation 2y = 4. We've successfully simplified the equation, bringing us one step closer to solving for 'y'. Simplification is not just about finding the answer; it's about making the problem more manageable. By breaking down complex expressions into simpler ones, we can reduce the chance of errors and gain a clearer understanding of the problem. So, with the equation simplified to 2y = 4, we're now ready for the final step: isolating 'y'.

4. Isolating 'y'

The final step in solving for 'y' is to isolate 'y' on one side of the equation. We have the simplified equation 2y = 4. To isolate 'y', we need to undo the multiplication by 2. We do this by dividing both sides of the equation by 2. So, (2y) / 2 = 4 / 2. Isolating a variable is a key technique in algebra. It involves performing operations on both sides of the equation to get the variable by itself. This allows us to determine the value of the variable. Think of it as peeling away the layers of an onion – we're gradually removing everything around 'y' until we reveal its true value. Dividing both sides of the equation by the same number maintains the equality. This is a fundamental principle of algebra. If we only divided one side, we would change the relationship between the two sides and get an incorrect answer. Performing the division, we get y = 2. We've successfully isolated 'y' and found its value. This means that when x = 12, the value of 'y' in the equation 2y = 10 - (1/2)x is 2. Isolating a variable requires careful attention to detail and a clear understanding of algebraic principles. It's like conducting a scientific experiment – we need to follow the procedure precisely to get accurate results. Now that we've found the value of 'y', let's summarize our steps and see the complete solution.

5. Conclusion: The Value of 'y'

In conclusion, by following the steps of substitution, simplification, and isolation, we've successfully evaluated the equation 2y = 10 - (1/2)x when x = 12. We found that the value of y is 2. This process demonstrates the power of algebra in solving real-world problems. We started with a general equation, plugged in a specific value for 'x', and solved for 'y'. This is a fundamental skill in mathematics and many other fields. Remember, the key to solving equations is to break them down into smaller, manageable steps. Start by understanding the equation, substitute the given values, simplify using the order of operations, and isolate the variable you're solving for. Practice is essential for mastering these techniques. The more equations you solve, the more comfortable and confident you'll become. Think of it as learning a new language – the more you practice, the more fluent you'll become. Equations are not just abstract mathematical concepts; they are tools for understanding and describing the world around us. From physics and engineering to economics and finance, equations are used to model and solve a wide range of problems. So, the ability to solve equations is a valuable skill that can open doors to many opportunities. We hope this step-by-step guide has helped you understand how to evaluate equations. Keep practicing, and you'll be solving equations like a pro in no time! For further learning and practice on algebra and equation solving, you can explore resources like Khan Academy's Algebra I section.