Solving 2x-y=24 And X+y=6 A Step-by-Step Guide

Hey guys! Math can sometimes feel like trying to untangle a really messy knot, right? But don't worry, because today we're going to break down a common type of math problem: solving a system of linear equations. Specifically, we're going to tackle the equations 2x - y = 24 and x + y = 6. Think of it like this: we have two clues (our equations) and two unknowns (x and y), and our mission is to figure out what those unknowns are. We'll walk through it step-by-step, so even if you're feeling a bit rusty with your algebra, you'll be able to follow along. Let's dive in and make those equations our friends!

Understanding Systems of Equations

Before we jump into solving our specific problem, let's zoom out for a second and talk about what a system of equations actually is. At its heart, a system of equations is just a set of two or more equations that share the same variables. In our case, we have two equations, and both of them involve the variables x and y. The whole point of solving a system of equations is to find the values for those variables that make all the equations in the system true at the same time. It's like finding the perfect combination that unlocks a secret code. There are several methods we can use to solve these systems, such as substitution, elimination, and graphing. Each method has its own strengths, and the best one to use often depends on the specific equations you're dealing with. For instance, if one of the equations already has a variable isolated (like y = something), the substitution method might be a great choice. On the other hand, if the coefficients of one of the variables are opposites (like +y and -y), the elimination method might be quicker. Understanding these different approaches gives you a powerful toolkit for tackling a wide range of problems. Recognizing patterns and choosing the most efficient method is a key skill in algebra, and it's something that will become more natural with practice. Remember, the goal is always the same: to find the values that satisfy every equation in the system simultaneously. It's like making sure every piece of a puzzle fits perfectly into place.

Method 1: The Elimination Method

Okay, let's get our hands dirty with the first method: the elimination method. This is a super useful technique, especially when you notice that the coefficients of one of the variables are either the same or opposites. In our problem, we have the equations 2x - y = 24 and x + y = 6. Notice anything interesting about the y terms? That's right! We have a -y in the first equation and a +y in the second equation. This is perfect for elimination! The basic idea behind elimination is to add the equations together in a way that cancels out one of the variables. In this case, if we simply add the two equations together, the y terms will disappear: (2x - y) + (x + y) = 24 + 6. This simplifies to 3x = 30. See how the y magically vanished? Now we have a much simpler equation with only one variable. To solve for x, we just divide both sides by 3: x = 10. Awesome! We've found the value of x. But we're not done yet – we still need to find y. This is where the second part of the elimination method comes in. Now that we know x, we can substitute it back into either of the original equations to solve for y. Let's use the second equation, x + y = 6, since it looks a bit simpler. Substituting x = 10, we get 10 + y = 6. To isolate y, we subtract 10 from both sides: y = -4. And there you have it! We've found both x and y: x = 10 and y = -4. To be absolutely sure we've got the right answer, we can plug these values back into both of the original equations to check. If both equations hold true, we know we're golden. The elimination method is a powerful tool because it allows us to systematically reduce a system of two equations into a single, solvable equation. It's like a strategic way to chip away at the problem until we reveal the solution.

Method 2: The Substitution Method

Alright, let's explore another method for solving this system: the substitution method. This approach is particularly handy when one of the equations can be easily rearranged to isolate a single variable. Looking at our equations, 2x - y = 24 and x + y = 6, the second equation, x + y = 6, seems like a good candidate for this. We can quickly isolate y by subtracting x from both sides: y = 6 - x. Now we have an expression for y in terms of x. This is the key to the substitution method – we're going to substitute this expression into the other equation. We take our expression for y, which is 6 - x, and plug it into the first equation, 2x - y = 24, wherever we see y. This gives us 2x - (6 - x) = 24. Be careful with those parentheses! They're super important because we need to distribute the negative sign. Expanding the equation, we get 2x - 6 + x = 24. Now we can combine like terms: 3x - 6 = 24. Next, we add 6 to both sides: 3x = 30. And finally, we divide both sides by 3: x = 10. Hey, that looks familiar! We got the same value for x as we did using the elimination method. This is a good sign that we're on the right track. Now that we know x = 10, we can substitute it back into our expression for y: y = 6 - x = 6 - 10 = -4. Boom! We've found y using the substitution method, and it matches our previous answer: y = -4. Just like with the elimination method, it's always a good idea to check our solution by plugging the values of x and y back into the original equations. This helps us catch any sneaky errors we might have made along the way. The substitution method is a bit like solving a puzzle by replacing one piece with another that fits perfectly. It's a powerful technique that can be especially useful when dealing with more complex systems of equations.

Verifying the Solution

So, we've crunched the numbers using both the elimination and substitution methods, and we've arrived at the solution x = 10 and y = -4. But before we declare victory and move on, it's crucial to verify our solution. Think of this as the final boss battle in our math quest – it's our chance to make absolutely sure we've slayed the problem. The best way to verify our solution is to plug the values of x and y back into the original equations and see if they hold true. Let's start with the first equation: 2x - y = 24. Substituting x = 10 and y = -4, we get 2(10) - (-4) = 20 + 4 = 24. Awesome! The first equation checks out. Now let's move on to the second equation: x + y = 6. Substituting x = 10 and y = -4, we get 10 + (-4) = 6. Fantastic! The second equation also holds true. Since our solution satisfies both equations, we can confidently say that x = 10 and y = -4 is indeed the correct answer. Verifying our solution is a fundamental step in problem-solving, not just in math, but in many areas of life. It's like double-checking your work before submitting a project or testing a recipe before serving it to guests. It gives us peace of mind and helps us avoid costly mistakes. By taking the time to verify, we solidify our understanding of the problem and reinforce the concepts we've learned. So, always remember to verify your solutions – it's the mark of a true math master!

Why This Matters: Real-World Applications

Okay, we've successfully solved the system of equations 2x - y = 24 and x + y = 6, but you might be thinking, "Okay, that's cool, but when am I ever going to use this in real life?" That's a totally valid question! The truth is, systems of equations pop up in all sorts of unexpected places. They're not just abstract math problems; they're powerful tools for modeling and solving real-world situations. Think about it: many real-world problems involve multiple variables that are related to each other in some way. Systems of equations provide a framework for expressing these relationships mathematically and finding solutions. For example, imagine you're running a business and you need to figure out how to price your products to maximize profit. You might have one equation that represents your revenue (based on price and quantity sold) and another equation that represents your costs. Solving this system of equations could tell you the optimal price to charge. Or, let's say you're planning a road trip and you want to know how long it will take to reach your destination. You might have one equation that relates distance, speed, and time, and another equation that represents your budget for gas. Solving this system could help you plan your trip efficiently. Systems of equations are also used extensively in science and engineering. They can be used to model electrical circuits, chemical reactions, and even the motion of planets. In economics, they're used to analyze supply and demand, and in computer science, they're used in areas like computer graphics and optimization. The specific examples could go on and on, but the key takeaway is that systems of equations are a fundamental tool for anyone who wants to analyze and solve problems involving multiple variables. By mastering these techniques, you're not just learning math; you're developing a valuable skill that can be applied in countless real-world situations. So, the next time you encounter a problem that seems complex and multifaceted, remember that systems of equations might just be the key to unlocking the solution.

Conclusion

Alright, guys, we've reached the end of our journey through the world of systems of equations! We started with the problem of solving 2x - y = 24 and x + y = 6, and we've explored not one, but two different methods for finding the solution: the elimination method and the substitution method. We saw how the elimination method works by cleverly adding the equations together to eliminate one variable, making the problem much simpler to solve. Then, we dove into the substitution method, where we isolated one variable and substituted its expression into the other equation, achieving the same result. We also emphasized the crucial step of verifying our solution, plugging our values back into the original equations to ensure everything checks out. But we didn't stop there! We also discussed why this all matters, exploring the many real-world applications of systems of equations, from business and finance to science and engineering. Systems of equations are a powerful tool for modeling and solving problems involving multiple variables, and mastering these techniques can open doors to a wide range of possibilities. So, what's the big takeaway? Solving systems of equations isn't just about crunching numbers; it's about developing problem-solving skills, thinking strategically, and understanding how math connects to the world around us. Whether you're balancing your budget, planning a trip, or designing a bridge, the ability to work with systems of equations will serve you well. Keep practicing, keep exploring, and remember that math can be an exciting and rewarding adventure! And most importantly, don't be afraid to ask for help when you need it. We're all in this together, and the more we learn from each other, the better we'll become at tackling any math challenge that comes our way. So, go forth and conquer those equations! You've got this!