Find The Solution: System Of Linear Equations
Hey math enthusiasts! Let's dive into a classic problem: finding the solution to a system of linear equations. We're going to take a look at how to find the ordered pair (x, y) that satisfies both equations. This is a fundamental concept in algebra, and understanding it opens doors to solving more complex problems. We'll break down the process step by step, making it easy to grasp. The question we're tackling is: which ordered pair is a solution to the given system of linear equations. Let's start this journey, shall we? This is not just about getting the right answer, it's about understanding the underlying concept. So, grab your pencils, and let's begin. We're not just going to find the answer; we're going to understand why that answer is correct. This exploration is designed to empower you with the tools and the knowledge to confidently solve these types of problems, so you'll be ready for any equation that comes your way. I hope you're as excited as I am to jump into the solution.
Understanding the Problem
Alright, let's get down to brass tacks. Our main task is to identify the ordered pair that works for both equations. Remember that an ordered pair, like (x, y), represents a point on a graph. A solution to a system of linear equations is the point where the lines represented by the equations intersect. Think of it as the spot where both equations are true at the same time. It's the magic coordinate that satisfies both equations simultaneously. In the system of equations, it is given that: $2x + 3y = 6$ and $-3x + 5y = 10$. We're provided with four possible solutions, which will be our choices: A. (0, 2), B. (2, 0), C. (3, 2), and D. (2, 3). The trick is to plug in the x and y values from each ordered pair into both equations and see which one makes both equations true. This is like a game of verification, we are going to test each option until we find the one that works. In short, this method is called substitution. It is useful in these types of scenarios, so the main idea is to replace the variables in the equations to verify which one satisfies both. Let's get started with our first choice and begin the process. By the end of this, you'll be a pro at solving systems of linear equations!
Testing the Options
Now, let's roll up our sleeves and test the options. We're going to methodically check each ordered pair, substituting the x and y values into both equations. This step-by-step process will show you exactly how to find the correct answer and why the other options don't quite fit. Let's start with option A, which is (0, 2). This means x = 0 and y = 2. We will plug these values into both equations. For the first equation, $2x + 3y = 6$, substituting x = 0 and y = 2, we get: $2(0) + 3(2) = 6$. This simplifies to $0 + 6 = 6$, which is true. So far, so good! Now, let's check the second equation: $-3x + 5y = 10$. Substituting x = 0 and y = 2, we have: $-3(0) + 5(2) = 10$. This simplifies to $0 + 10 = 10$, which is also true. Since (0, 2) satisfies both equations, this is our solution. But, let's go ahead and check the other options just to be sure. It is good to go the extra mile and verify, just in case. Let's examine option B (2, 0). Here, x = 2 and y = 0. Plugging these values into the first equation: $2(2) + 3(0) = 6$. This simplifies to $4 + 0 = 6$, which is false. Since the first equation isn't true, we know that (2, 0) is not a solution. This helps to confirm our previous answer. Now, on to option C (3, 2). Here, x = 3 and y = 2. For the first equation: $2(3) + 3(2) = 6$, which simplifies to $6 + 6 = 6$, again false. So, (3, 2) is also not a solution. Lastly, let's check option D (2, 3). Here, x = 2 and y = 3. For the first equation: $2(2) + 3(3) = 6$, which simplifies to $4 + 9 = 6$, also false. So, (2, 3) is not a solution either.
Conclusion: The Correct Answer
After carefully testing each option, we have found our winner. Option A, (0, 2), is the solution to the system of linear equations. This means that when x = 0 and y = 2, both equations in the system are true. It's the single point where the two lines represented by the equations intersect on a graph. Now that we have gone through the whole process, you've seen how to find the solution through substitution. Remember that this method involves taking the values of x and y from the ordered pairs and replacing them in both equations. If an ordered pair satisfies both equations, then it's the solution. If even one equation is false, then that ordered pair is not the solution. This approach is straightforward and it's a fundamental skill for solving problems in algebra and beyond. Understanding the core concepts makes you a more confident problem solver. Keep practicing, keep experimenting, and you will surely become a master of solving systems of linear equations. Always remember to double-check your work, and don't be afraid to go back and review the steps. This process is all about precision and understanding, and you're now well-equipped to handle these types of problems!