Is (4, -3) A Solution? Checking Equations

Hey math enthusiasts! Today, we're diving into a super important concept: figuring out if an ordered pair is a solution to a given equation. Specifically, we're going to tackle the question: Is the ordered pair (4, -3) a solution to the equation 3x + 2y = 6? Don't worry, it's not as scary as it sounds! It's actually a pretty straightforward process, and once you get the hang of it, you'll be solving these problems like a pro. So, grab your pencils (or keyboards!), and let's get started. Understanding this concept is crucial because it lays the groundwork for more complex algebraic topics. Being able to quickly and accurately determine if a point lies on a line (represented by the equation) is a fundamental skill. Plus, it's super useful in real-life scenarios, even if you don't realize it! Think about it: anytime you're trying to find a point that satisfies certain conditions, you're essentially using the same principles. This process is also the foundation for graphing linear equations, solving systems of equations, and understanding the relationship between algebra and geometry. The ability to verify solutions is a cornerstone of mathematical problem-solving, ensuring accuracy and building confidence in your abilities. The skill is essential for anyone looking to excel in mathematics, from high school students to professionals in fields that rely on mathematical modeling and analysis.

Understanding Ordered Pairs and Equations

Alright, before we jump into the problem, let's make sure we're all on the same page. First off, what exactly is an ordered pair? Well, it's simply a pair of numbers, written in a specific order, and enclosed in parentheses, like this: (x, y). In our case, we have the ordered pair (4, -3). The first number, 4, represents the x-coordinate, and the second number, -3, represents the y-coordinate. Think of these coordinates as the location of a point on a graph. The equation 3x + 2y = 6, on the other hand, represents a linear equation. This equation describes a straight line when graphed. The equation expresses the relationship between x and y. A solution to an equation is any ordered pair (x, y) that makes the equation true. In other words, if you plug in the x and y values from the ordered pair into the equation, and the equation holds true (the left side equals the right side), then the ordered pair is a solution. Otherwise, it isn't. This means that if we substitute x = 4 and y = -3 into the equation, and it results in a true statement, then the ordered pair (4, -3) lies on the line represented by the equation. If the statement is false, the ordered pair does not lie on the line. Understanding the role of ordered pairs and equations is fundamental to visualizing mathematical concepts and solving problems. The combination of the equation and the ordered pair is a basic element in understanding the coordinate plane. In our quest to solve the problem, the importance of each part of the question is relevant to solving it, helping in creating a solid foundation for subsequent mathematical concepts.

Step-by-Step: Checking the Solution

Now, let's get down to business and actually solve the problem. Here's how we can determine if (4, -3) is a solution to 3x + 2y = 6. First, we need to substitute the values of x and y from the ordered pair into the equation. In our ordered pair (4, -3), x = 4 and y = -3. So, we replace x with 4 and y with -3 in the equation 3x + 2y = 6. This gives us: 3(4) + 2(-3) = 6. Next, we need to simplify this equation. Follow the order of operations (PEMDAS/BODMAS), which tells us to handle multiplication before addition or subtraction. So, we multiply 3 by 4, which gives us 12, and we multiply 2 by -3, which gives us -6. The equation now looks like this: 12 - 6 = 6. Finally, we perform the subtraction. 12 - 6 equals 6. This gives us 6 = 6. This is a true statement! The equation is balanced, the left side equals the right side. Because this statement is true, it indicates that the ordered pair (4, -3) satisfies the equation. We can say that the ordered pair (4, -3) lies on the line represented by the equation 3x + 2y = 6. This systematic approach is vital, as it eliminates any doubt in the solution, making it easy to apply in a diverse set of equations. This exercise will help with mastering the technique of substituting values into equations, an essential skill for further mathematical studies. The step-by-step process ensures you always arrive at the right answer, reinforcing the concept. Remember, practice makes perfect! By carefully following these steps, you'll not only correctly solve the problem but also gain a deeper understanding of the underlying mathematical principles. This process is designed to be easy to follow. The method ensures the user has a proper, and concrete understanding of the question.

Conclusion: Is (4, -3) a Solution?

So, guys, the answer is a resounding YES! The ordered pair (4, -3) is a solution to the equation 3x + 2y = 6. When we plugged in the values of x and y, the equation held true. This confirms that the point represented by the ordered pair lies on the line defined by the equation. This is a simple example of how we can verify solutions to equations, a fundamental concept in algebra. To recap, we've learned how to substitute the x and y values from an ordered pair into an equation, simplify the equation using the order of operations, and determine whether the ordered pair is a solution based on whether the resulting statement is true or false. This approach is not only applicable to linear equations but can also be used to check solutions for other types of equations, such as quadratic equations or even more complex mathematical models. This skill is very useful in everyday life, from financial planning to physics calculations. Always remember the process of substitution, simplification, and verification. Keeping these things in mind, you'll be able to solve more complex problems, solidifying your mathematical understanding. This reinforces the importance of the skill, extending its value beyond the classroom. This skill is important and will help build confidence in solving mathematical problems.

Practice Makes Perfect! More Examples

Alright, you've grasped the concept, but the only way to truly master it is to practice! Here are a few more examples for you to try. For each ordered pair and equation, determine whether the ordered pair is a solution. Remember to follow the same steps: substitute, simplify, and check. Example 1: Equation: 2x - y = 5. Ordered pair: (3, 1). Example 2: Equation: y = x + 2. Ordered pair: (0, 2). Example 3: Equation: x + y = 0. Ordered pair: (-2, 2). Go ahead and give these a shot. The more you practice, the more comfortable and confident you'll become. Remember to use the order of operations and double-check your calculations. The solutions are included below, so you can check your work once you're done. Don't be discouraged if you don't get them all right away. It takes practice! Each problem presents a unique context, helping you in perfecting and reinforcing the skills. Practicing with different equations and ordered pairs builds confidence and helps you to apply the process in various scenarios. This extra section reinforces what has already been learnt, increasing the user's skill. This also allows the user to identify potential difficulties and resolve them with the solutions. By engaging with these additional examples, you're actively solidifying your understanding and preparing yourself for more complex mathematical challenges. You'll become more efficient at solving these types of problems. The ability to identify and understand solutions will become second nature to you.

Solutions to the Practice Problems

Let's check your work! Here are the solutions to the practice problems: For Example 1: Equation: 2x - y = 5. Ordered pair: (3, 1). Solution: Yes, (3, 1) is a solution. Substituting x = 3 and y = 1 into the equation, we get 2(3) - 1 = 5, which simplifies to 6 - 1 = 5, and further simplifies to 5 = 5. This is a true statement, indicating the ordered pair is a solution. For Example 2: Equation: y = x + 2. Ordered pair: (0, 2). Solution: Yes, (0, 2) is a solution. Substituting x = 0 and y = 2 into the equation, we get 2 = 0 + 2, which simplifies to 2 = 2. This is a true statement, confirming the ordered pair is a solution. For Example 3: Equation: x + y = 0. Ordered pair: (-2, 2). Solution: No, (-2, 2) is not a solution. Substituting x = -2 and y = 2 into the equation, we get -2 + 2 = 0, which simplifies to 0 = 0. This is not a true statement, meaning the ordered pair is not a solution. Compare your answers with these solutions. Take your time to understand any discrepancies and learn from them. This exercise reinforces the importance of checking your calculations. Also, take note of the cases where the ordered pair doesn't satisfy the equation, which is just as important as the ones that do. Remember, the goal isn't just to get the right answer but to understand the underlying concepts. Reviewing the solutions can help you pinpoint areas where you need more practice. By using the solutions to measure your results, you're developing crucial problem-solving skills that will be valuable in all your math endeavors. Analyzing the examples will help in mastering the skill and becoming proficient in verifying solutions to equations. These solutions aim to help you improve with the subject, while also building up your confidence.