Solving Linear Equations The Elimination Method

Hey guys! Ever stumbled upon a system of linear equations and felt a bit lost? Don't worry; you're not alone! Linear equations are a fundamental part of mathematics, and solving them is a crucial skill. In this article, we'll break down how to tackle these problems, using a real example to guide you through the process. So, let's dive in and make those equations a little less intimidating!

Understanding Systems of Linear Equations

Before we jump into the solution, let's quickly recap what a system of linear equations actually is. Systems of linear equations involve two or more linear equations that share the same variables. The goal is to find the values of these variables that satisfy all equations simultaneously. Think of it like finding the perfect meeting point for multiple lines on a graph. Each equation represents a line, and the solution is where these lines intersect. There are several methods to solve these systems, including substitution, elimination, and graphing. We'll focus on the elimination method in this example, as it's particularly efficient for the problem at hand. Remember, a linear equation is simply an equation where the highest power of the variable is 1. This means we're dealing with straight lines, not curves or anything more complex. Mastering systems of linear equations is essential because they pop up in various real-world scenarios, from calculating costs and profits in business to modeling relationships in science and engineering. Now that we have a solid grasp of what we're dealing with, let's move on to the specific problem we're going to solve. We'll take a look at the equations themselves and start thinking about how to approach them systematically. Understanding the underlying principles makes the entire process much smoother and less daunting, so don't skip this important foundational step! We are trying to find the values of 'x' and 'y' that make both equations true at the same time. This point, (x, y), represents the intersection of the two lines if we were to graph them. Solving linear equations might seem complex, but breaking it down into manageable steps makes it much easier. Each method has its strengths, and choosing the right one can save you time and effort. The elimination method, also known as the addition method, is particularly useful when the coefficients of one variable are opposites or can easily be made opposites. This allows us to eliminate one variable by adding the equations together, simplifying the system. In the real world, these systems can model anything from supply and demand curves to mixing solutions in chemistry. For instance, imagine you're running a lemonade stand. You have two different recipes, each using varying amounts of lemons and sugar. By setting up a system of equations, you can determine the optimal amounts of each ingredient to maximize your profit or achieve a specific flavor profile. This practical application highlights the importance of understanding these concepts.

The Problem: A System of Two Equations

Okay, let's get to the heart of the matter! We're faced with the following system of linear equations:

2x + 3y = 3
7x - 3y = 24

Our mission, should we choose to accept it (and we do!), is to find the values of x and y that satisfy both of these equations simultaneously. Notice anything interesting about these equations? Look closely at the y terms. In the first equation, we have +3y, and in the second, we have -3y. This is a huge clue that the elimination method is going to be our best friend here. The coefficients of y are already opposites, which means we're halfway to solving this problem! Before we dive into the elimination process, let's just take a moment to appreciate the elegance of this setup. The beauty of mathematics often lies in recognizing these patterns and using them to our advantage. Now, why is solving this system important? Well, imagine these equations represent different constraints or conditions in a real-world scenario. Maybe they describe the amount of resources available and the requirements for a certain project. Finding the solution (x and y) would then tell us the optimal way to use those resources. This is a simplified example, of course, but it illustrates the power of linear equations in modeling and solving practical problems. So, with our detective hats on, let's move on to the next step: eliminating the y variable and getting closer to our solution. We need to understand what each part of the equation represents. The numbers in front of the variables (x and y) are called coefficients. They tell us how much of that variable we have. The numbers on the right side of the equal sign are constants. They represent the total value or the target we're trying to reach. For example, in the first equation, 2x + 3y = 3, the coefficient 2 tells us we have two xs, the coefficient 3 tells us we have three ys, and the constant 3 tells us that the sum of two xs and three ys must equal 3. Understanding this structure is crucial for manipulating equations and solving for the unknowns. In the second equation, 7x - 3y = 24, the negative sign in front of the 3y indicates that we are subtracting three ys. This is another important detail to pay attention to, as it affects how we manipulate the equation. Notice the strategic advantage of having opposite coefficients for y. This setup allows us to easily eliminate y by adding the equations, a core technique in solving systems of equations. Without this, we would need to perform additional steps to create opposite coefficients. This demonstrates the importance of observation and pattern recognition in problem-solving.

Step-by-Step Solution Using Elimination

Alright, let's get our hands dirty and solve this thing! As we've already discussed, the elimination method is perfect for this scenario because the y coefficients are opposites. Here's how it works:

1. Add the equations together: This is the key step. We're going to add the left-hand sides of the equations and the right-hand sides separately.

   (2x + 3y) + (7x - 3y) = 3 + 24
   

2. Simplify: Now, let's combine like terms. Notice what happens to the y terms – they cancel each other out!

   2x + 7x + 3y - 3y = 27
   9x = 27
   

3. Solve for x: We're left with a simple equation with just one variable. Divide both sides by 9 to isolate x.

   x = 27 / 9
   x = 3
   

   Eureka! We've found the value of x. Now, we need to find y. This is where the next step comes in: substitution.

4. Substitute x into one of the original equations: We can choose either equation, but let's go with the first one (2x + 3y = 3) for simplicity.

   2(3) + 3y = 3
   

5. Solve for y: Now, it's just a matter of algebra.

   6 + 3y = 3
   3y = 3 - 6
   3y = -3
   y = -3 / 3
   y = -1
   

   Fantastic! We've found the value of y as well.

So, our solution is x = 3 and y = -1. But we're not done yet! It's always a good idea to check our work.

Verifying the Solution

Okay, we've found our solution: x = 3 and y = -1. But how do we know if we're right? The best way to be sure is to check our solution in both of the original equations. This is a crucial step that many people skip, but it can save you from making mistakes. It's like double-checking your work in any other task – it ensures accuracy and gives you confidence in your answer. Let's plug our values into the first equation:

2x + 3y = 3
2(3) + 3(-1) = 3
6 - 3 = 3
3 = 3

Great! It works for the first equation. Now, let's try the second equation:

7x - 3y = 24
7(3) - 3(-1) = 24
21 + 3 = 24
24 = 24

Awesome! Our solution works for both equations. This confirms that we've found the correct values for x and y. If our solution didn't work in either equation, we would know that we made a mistake somewhere and would need to go back and review our steps. This verification process is not just about getting the right answer; it's about developing good mathematical habits and building a deeper understanding of the problem. By checking our solution, we're reinforcing the connection between the equations and the values that satisfy them. We are verifying is crucial in mathematics because it confirms that our solution is consistent with the given conditions. If our solution fails to satisfy one or both equations, it indicates an error in our calculations, which prompts us to review our steps and correct any mistakes. This process not only ensures the accuracy of our answer but also enhances our problem-solving skills and understanding of the underlying mathematical principles. Furthermore, checking our solution is a fundamental aspect of mathematical rigor and precision. It demonstrates a commitment to thoroughness and attention to detail, which are essential qualities in any scientific or technical endeavor. In practical applications, verifying solutions can prevent costly errors and ensure the reliability of our results. For instance, in engineering, an incorrect solution to a system of equations could lead to structural failures or other serious consequences. Therefore, adopting a systematic approach to problem-solving, including the step of verification, is crucial for success in mathematics and related fields.

The Answer: Option A (3, -1)

So, after all that hard work, what's the final answer? We found that x = 3 and y = -1. Looking at the options provided, this corresponds to Option A: (3, -1). We did it! We successfully solved the system of linear equations. Give yourselves a pat on the back, guys! You've navigated through the steps of the elimination method, found the values of x and y, and even verified your solution. This is a fantastic accomplishment, and you should feel proud of your progress. But more importantly than just getting the right answer, you've gained valuable skills and insights into how to approach mathematical problems systematically. Remember, problem-solving is not just about memorizing formulas and procedures; it's about understanding the underlying concepts and applying them creatively. The skills you've honed in this exercise, such as observation, logical reasoning, and attention to detail, will serve you well in many areas of life, both inside and outside the classroom. As we wrap up this example, let's take a moment to reflect on the key takeaways. We learned how to identify a system of linear equations, how to choose the appropriate method (elimination in this case), how to execute the steps of the method, and how to verify our solution. These are fundamental skills that you can build upon as you continue your mathematical journey. And remember, practice makes perfect! The more you work with these concepts, the more confident and proficient you'll become. So, don't be afraid to tackle more problems and explore different types of systems of equations. The world of mathematics is vast and fascinating, and there's always something new to discover. So, keep exploring, keep learning, and keep having fun with math!

Conclusion

And there you have it! Solving systems of linear equations doesn't have to be a mystery. By understanding the concepts, following a systematic approach, and double-checking your work, you can conquer these problems with confidence. Remember, the elimination method is just one tool in your mathematical toolbox. There are other methods, like substitution and graphing, that can be useful in different situations. The key is to choose the method that best suits the problem at hand and to practice regularly. Keep exploring, keep learning, and keep solving! You've got this, guys!