Solving For 'y': A Step-by-Step Guide To Algebraic Equations
Hey there, math enthusiasts! Today, we're diving into the world of algebra to figure out how to solve for a variable, specifically 'y,' in a linear equation. We'll be breaking down the process step by step, making sure it's easy to understand. So, grab your pencils and let's get started. We're going to address the equation: 300 + 3y = 450. Don't worry if this looks intimidating; we'll tackle it bit by bit. Understanding how to solve for a variable is a fundamental skill in mathematics, opening doors to more complex problem-solving down the line. This process is not just about finding a number; it's about understanding the relationships between numbers and how they interact within equations. This knowledge is applicable across various fields, from science and engineering to economics and computer science. This knowledge is really important, and solving for 'y' or any variable is a crucial skill. This exercise is a building block for more complex mathematical concepts. The goal is to confidently manipulate equations to isolate the unknown variable and discover its value.
We'll start with the basics, ensuring you have a strong foundation. Then, we'll gradually move towards solving the equation. Our primary objective is to isolate 'y' on one side of the equation, which means we need to get 'y' by itself. It's like trying to separate one ingredient from a complex recipe. The methods we'll use involve applying the properties of equality, ensuring that any operation performed on one side of the equation is also done on the other to maintain balance. This ensures that the equation remains valid throughout the steps. Ready to begin?
Step 1: Setting the Stage - Understanding the Equation
Alright, guys, let's start with the basics! Our equation is 300 + 3y = 450. In this equation, 'y' is the variable we want to find. The number '3' next to 'y' means it's multiplied by 'y' (3 * y), and '300' and '450' are constants, which are just regular numbers. Think of it like this: you have 300, and you add three times an unknown amount to get to 450. Our job is to figure out what that unknown amount is! We need to isolate 'y' on one side of the equation and get it by itself. First, identify the different components: the constant terms (300 and 450), the coefficient of the variable (3), and the variable itself ('y'). Understanding these parts is key to knowing how to manipulate the equation effectively. This is the foundation of the solution.
To solve the equation, we'll use the properties of equality to maintain the balance. This means that whatever we do on one side of the equation, we must also do on the other side to keep it true. This is just like a balanced scale; if you add or remove weight from one side, you must do the same to the other to keep it level. This principle is crucial to avoid changing the equation's value and losing its accuracy. Remember this basic idea, and you'll be in good shape for algebra.
Let's proceed to the next step, where we will start to isolate 'y'. We will use subtraction to remove the constant on one side of the equation.
Step 2: Isolating 'y' - Subtracting the Constant
Here's where the magic happens! Our main goal is to get 'y' alone. The first step is to get rid of the '300' on the left side of the equation. We can do this by subtracting 300 from both sides of the equation. Remember, whatever we do to one side, we must do to the other. So, we'll subtract 300 from both sides.
Original equation: 300 + 3y = 450
Subtract 300 from both sides: 300 + 3y - 300 = 450 - 300
Now, simplify: 3y = 150
See, it's not so bad, right? We've moved the constant term from one side to the other by subtracting it from both sides. Subtracting 300 from the left side cancels out the 300, leaving us with just 3y. On the right side, 450 - 300 gives us 150. This step simplifies the equation, bringing us closer to finding the value of 'y'. This operation is based on the principle of equality, which maintains the balance of the equation. We subtract 300 on both sides to maintain equivalence.
Notice how we systematically moved the numbers away from the 'y' term to isolate it. This is a core strategy in algebra and will be used in more advanced equations as well. We're now ready to solve for the last step: division!
Step 3: Solving for 'y' - The Final Division
We're in the home stretch now! We've got the equation 3y = 150, and we want to find the value of 'y.' Currently, 'y' is being multiplied by 3. To get 'y' by itself, we need to do the opposite operation: divide both sides of the equation by 3.
Equation: 3y = 150
Divide both sides by 3: (3y) / 3 = 150 / 3
Simplify: y = 50
And there you have it! We have successfully solved for 'y'. We divided both sides by 3, which cancels out the 3 on the left side, leaving us with just 'y'. On the right side, 150 divided by 3 equals 50. Therefore, y = 50. See, it wasn't as difficult as it looked, right?
This final division step is crucial. It completes the process of isolating the variable. Division by 3 isolates the variable 'y', making its value directly apparent. This last step requires a good understanding of arithmetic operations and properties. This demonstrates a complete understanding of the equation. Always double-check your answer by substituting it back into the original equation to confirm its accuracy.
Step 4: Verification and Understanding the Solution
Now, let's verify our answer. We found that y = 50. To make sure we're right, we'll substitute 50 back into the original equation to see if it works: 300 + 3y = 450.
Substitute y = 50: 300 + 3(50) = 450
Calculate: 300 + 150 = 450
Simplify: 450 = 450
Since both sides of the equation are equal, our answer, y = 50, is correct! This means that the value we found for 'y' satisfies the original equation. This process, known as verification, is an essential step in algebra and helps prevent errors. It also boosts confidence in your problem-solving abilities.
Moreover, let's understand what this means in practical terms. Imagine you started with 300 items, and after adding three groups of 'y' items, you ended up with 450 items. Our solution tells us that each of those 'y' groups contained 50 items. Understanding this context is critical for applying algebraic skills in the real world. Math becomes much more accessible when you relate it to real-life situations.
Conclusion: Mastering the Equation
So, there you have it, guys! We've walked through solving a simple algebraic equation step by step. You've learned how to isolate a variable, use the properties of equality, and check your answers. Solving for 'y' is a foundational skill in algebra. We've moved from a seemingly complex equation to a simple solution, proving that, with the right approach, any equation can be solved! Keep practicing, and you'll be solving equations like a pro in no time. Remember to break down complex problems into smaller, manageable steps. Always verify your answers to ensure accuracy. Math can be challenging, but with these simple steps, it becomes manageable. Keep practicing, and you'll ace it!
Keep in mind that this process forms the basis for solving far more complex equations. With a good understanding of these methods, you can build your skills and conquer many other mathematical problems. The key takeaway is understanding the steps and the properties of equality, which is fundamental in the study of algebra. Good luck, and keep solving!