Solving Equations: Step-by-Step Guide
Hey there, math enthusiasts! Let's dive into the world of equations and learn how to solve them like pros. We're going to break down the equation y + 3 = -y + 9, step by step, making sure everyone understands the process. Don't worry, it's easier than it looks! Equations are fundamental in mathematics, and mastering them opens doors to solving more complex problems. This guide will provide a clear and concise walkthrough, ensuring you grasp the core concepts. So, grab your pens, and let's get started on this exciting mathematical journey. We'll cover everything from the basic principles to the final solution, ensuring you're equipped with the knowledge to tackle similar problems with confidence. Understanding equations is not just about getting the right answer; it's about building a solid foundation in mathematics that will serve you well in various fields. Whether you're a student, a professional, or simply someone curious about math, this guide is designed to help you succeed. By the end, you'll be able to solve equations with ease and confidence. The first step to solving an equation is to isolate the variable. This means getting the variable, in this case 'y', by itself on one side of the equation. We'll achieve this by using some fundamental algebraic rules. Remember, the goal is to manipulate the equation in a way that keeps it balanced. Let's get started and see how we can get our answer.
Step-by-Step Solution
Alright, let's get down to business and solve the equation y + 3 = -y + 9. We'll go through each step slowly, ensuring you understand every move we make. Our goal is to isolate 'y'.
Step 1: Combine 'y' Terms
To begin, we want to get all the 'y' terms on one side of the equation. We can do this by adding 'y' to both sides. Adding 'y' to both sides keeps the equation balanced. So we have:
y + 3 + y = -y + 9 + y
This simplifies to:
2y + 3 = 9
See? We're already making progress. Now the equation has been simplified, making it easy to continue. This step is designed to group all the 'y' terms, setting us up for the next phase: isolating the variable itself. Always remember the golden rule: whatever you do to one side of the equation, you must do to the other to maintain balance.
Step 2: Isolate the 'y' Term
Now that we've grouped the 'y' terms, our next step is to isolate the term with 'y'. To do this, we need to get rid of the '+3' on the left side. We can do this by subtracting 3 from both sides of the equation. Remember, the goal is to get 'y' by itself. This step is crucial because it begins the process of reducing the equation to its simplest form, making it easier to solve. Here's what that looks like:
2y + 3 - 3 = 9 - 3
This simplifies to:
2y = 6
Awesome, we're getting closer to the solution!
Step 3: Solve for 'y'
Finally, we need to solve for 'y'. Right now, we have '2y = 6'. To get 'y' by itself, we need to divide both sides of the equation by 2. This isolates 'y' and gives us the value we're looking for. Here's the final step:
2y / 2 = 6 / 2
Which simplifies to:
y = 3
And there you have it! We have found the solution. By following these steps, you'll be able to solve for 'y' with any linear equation with ease. This methodical approach ensures you understand each step and build confidence. Keep practicing, and you'll find yourself solving equations without even thinking about it!
Answer Choice Analysis
Now that we've solved the equation and found that y = 3, let's look at the answer choices provided and see which one matches our solution.
- A. y = 1: This is incorrect. It doesn't match our calculated solution. The value does not satisfy the original equation, meaning when plugged into the equation y + 3 = -y + 9, the left side does not equal the right side.
- B. y = 3: This is correct. It matches our calculated solution. When we substitute y = 3 into the original equation, both sides are equal.
- C. y = 6: This is incorrect. The solution is not equal to 6. Substituting 6 for y will not result in a balanced equation. Always double-check your work by substituting the solution back into the original equation.
- D. y = 9: This is also incorrect. It's important to note that understanding each answer choice, even the incorrect ones, can reinforce your understanding of the problem-solving process. Analyzing incorrect options helps prevent common mistakes.
So, the correct answer is B. y = 3. Congratulations! You've successfully solved the equation. Keep up the fantastic work, and you'll become a pro in no time. Remember to always double-check your answers to make sure they make sense in the context of the problem. Good luck with your mathematical journey!
Conclusion
In this guide, we've walked through solving the equation y + 3 = -y + 9 step-by-step. We covered combining like terms, isolating the variable, and solving for 'y'. We also analyzed the answer choices to ensure a clear understanding of the solution. Solving equations is a fundamental skill in mathematics, and we hope this guide has boosted your confidence. Keep practicing, and you'll master this skill in no time. Never stop learning, and always challenge yourself. Remember that every step you take builds a strong foundation for future mathematical concepts. Now go out there and solve some equations!
