Solving -9x - 5y = -18 And 4x + 5y = 33 A Step-by-Step Guide

Hey guys! Ever stumbled upon a system of equations that looks like it's straight out of a math textbook and thought, "Where do I even begin?" Well, you're not alone! Systems of equations can seem intimidating at first, but once you understand the basic concepts and the methods to solve them, they become a piece of cake. Today, we're going to break down a specific system of equations: -9x - 5y = -18 and 4x + 5y = 33. We'll explore how to solve it step-by-step, discuss the underlying principles, and highlight why this is such a fundamental topic in mathematics. So, buckle up, grab your favorite beverage, and let's dive into the world of systems of equations!

Understanding Systems of Equations

Before we jump into solving our specific system, let's take a moment to understand what a system of equations actually is. At its core, a system of equations is a set of two or more equations that contain the same variables. The goal is to find the values of these variables that satisfy all equations simultaneously. Think of it like a puzzle where you have multiple pieces, and they all need to fit together perfectly. In our case, we have two equations with two variables, x and y. This is a common scenario, but systems can also involve more equations and more variables. The key concept here is that we're looking for a common solution – a set of values for x and y that makes both equations true.

Why are systems of equations so important? Well, they pop up everywhere in real-world applications. From calculating the optimal mix of ingredients in a recipe to determining the trajectory of a rocket, systems of equations provide a powerful tool for modeling and solving complex problems. They are the backbone of many scientific and engineering calculations, and understanding them is crucial for anyone pursuing careers in these fields. Plus, they are super useful for problem-solving in general! So, mastering systems of equations is not just about acing your math test; it's about developing a valuable skill that will serve you well in many aspects of life.

In our particular system, -9x - 5y = -18 and 4x + 5y = 33, we can immediately notice a key feature: the y terms have opposite coefficients. This is a huge hint that the elimination method might be the easiest way to solve this system, but we'll get to that in a bit. First, let's appreciate the beauty of these equations. Each one represents a line on a graph, and the solution to the system is the point where these lines intersect. Isn't that neat? This visual representation can be incredibly helpful for understanding what's going on, especially when dealing with more complex systems. So, remember, systems of equations are not just abstract symbols; they have a concrete geometric interpretation.

Choosing the Right Method: Elimination

Okay, now that we have a solid understanding of what systems of equations are and why they're important, let's talk about how to solve them. There are several methods available, each with its own strengths and weaknesses. The most common methods are substitution, elimination, and graphing. The best method to use often depends on the specific system of equations you're dealing with. In some cases, one method might be significantly easier than the others.

For our system, -9x - 5y = -18 and 4x + 5y = 33, the elimination method stands out as the most efficient approach. Why? Because the y terms have opposite coefficients (-5 and +5). This means that if we simply add the two equations together, the y terms will cancel out, leaving us with a single equation in x. This simplifies the problem dramatically, allowing us to solve for x quickly and easily. The elimination method is particularly effective when you notice coefficients that are the same or opposites, as it allows you to eliminate one variable and solve for the other.

Now, let's briefly consider why the other methods might not be as ideal in this case. The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. While this method works for any system of equations, it can be more cumbersome in this case because neither equation is particularly easy to solve for a single variable. We would have to deal with fractions and extra steps, which increases the risk of making a mistake. Graphing is another option, but it's generally less precise than algebraic methods, especially when the solutions are not integers. Graphing is useful for visualizing the system and understanding the concept of intersection points, but it's not the most reliable way to find exact solutions.

So, the elimination method is our weapon of choice for this particular system. It's like having a perfectly tailored tool for the job – it fits the situation perfectly and allows us to solve the problem efficiently. Remember, choosing the right method is a crucial skill in mathematics. It can save you time, reduce the chance of errors, and make the whole problem-solving process much smoother. We are selecting the best method to approach systems of equations in a structured way.

Step-by-Step Solution Using Elimination

Alright, let's get down to the nitty-gritty and solve our system of equations using the elimination method. We've already established that this is the most efficient approach, so now it's time to put it into action. Remember our system:

-9x - 5y = -18

4x + 5y = 33

The first step, as we discussed, is to add the two equations together. Notice how the -5y and +5y terms perfectly cancel each other out. This is the magic of the elimination method! When we add the equations, we get:

(-9x - 5y) + (4x + 5y) = -18 + 33

Now, let's simplify this equation. Combine the x terms and the constant terms:

-5x = 15

See how we've reduced the system to a single equation with just one variable? This is a huge step forward! Now, we can easily solve for x by dividing both sides of the equation by -5:

x = 15 / -5

x = -3

We've found the value of x! That's the first half of our solution. Now, we need to find the value of y. To do this, we can substitute the value of x we just found into either of the original equations. It doesn't matter which one you choose; you'll get the same value for y. Let's use the second equation, 4x + 5y = 33, because it looks a bit simpler:

4(-3) + 5y = 33

Now, simplify and solve for y:

-12 + 5y = 33

Add 12 to both sides:

5y = 45

Divide both sides by 5:

y = 9

And there you have it! We've found the value of y. So, the solution to our system of equations is x = -3 and y = 9. We can write this as an ordered pair: (-3, 9). This means that the point (-3, 9) is the intersection point of the two lines represented by our equations.

Verifying the Solution

Before we celebrate our victory, it's always a good idea to verify our solution. This is a crucial step in problem-solving, as it ensures that we haven't made any mistakes along the way. To verify our solution, we simply substitute the values of x and y we found back into both of the original equations. If the equations hold true, then we know our solution is correct.

Let's start with the first equation, -9x - 5y = -18. Substitute x = -3 and y = 9:

-9(-3) - 5(9) = -18

Simplify:

27 - 45 = -18

-18 = -18

The first equation holds true! That's a good sign. Now, let's check the second equation, 4x + 5y = 33:

4(-3) + 5(9) = 33

Simplify:

-12 + 45 = 33

33 = 33

The second equation also holds true! This confirms that our solution, x = -3 and y = 9, is indeed correct. We've successfully solved the system of equations and verified our answer. Give yourselves a pat on the back!

Verifying your solution is not just about checking for mistakes; it's also about building confidence in your problem-solving abilities. When you know you've taken the time to verify your answer, you can be sure that you've done the work correctly. This is a valuable habit to develop, not just in mathematics, but in any field where accuracy is important. Remember, a little extra effort in verification can save you from a lot of headaches down the road.

Real-World Applications and Why This Matters

We've successfully navigated the math and found our solution, but let's zoom out for a moment and consider why systems of equations matter in the real world. This isn't just about abstract math concepts; it's about tools that help us understand and solve real-world problems. Systems of equations are used in a surprisingly wide range of applications, from economics and engineering to computer science and even everyday life.

Think about economics, for example. Economists use systems of equations to model supply and demand, predict market trends, and analyze the impact of government policies. The intersection point of the supply and demand curves, which is the equilibrium price and quantity, is essentially the solution to a system of equations. In engineering, systems of equations are used to design structures, analyze circuits, and optimize processes. For instance, engineers might use systems of equations to calculate the forces acting on a bridge or to determine the flow rates in a network of pipes.

In computer science, systems of equations are used in areas like computer graphics, game development, and machine learning. For example, in computer graphics, systems of equations are used to transform and project 3D objects onto a 2D screen. In game development, they're used to simulate physics and interactions between objects. And in machine learning, they're used in algorithms for tasks like linear regression and classification.

Even in everyday life, we encounter situations that can be modeled using systems of equations. For instance, if you're planning a party and need to figure out how much food and drinks to buy, you might use a system of equations to balance your budget and the number of guests. Or, if you're trying to optimize your workout routine, you might use a system of equations to track your calorie intake and expenditure.

Understanding systems of equations not only strengthens your mathematical foundation but also equips you with a powerful tool for problem-solving in various contexts. It's about developing critical thinking skills and the ability to approach complex problems in a structured way. By mastering these concepts, you're opening doors to a wide range of opportunities and empowering yourself to tackle real-world challenges with confidence.

Conclusion: Mastering the Basics for Future Success

Well, guys, we've reached the end of our journey into solving the system of equations -9x - 5y = -18 and 4x + 5y = 33. We've not only found the solution (x = -3, y = 9), but we've also explored the underlying concepts, the importance of choosing the right method (elimination, in this case), the crucial step of verifying our solution, and the real-world applications of systems of equations. We've seen how seemingly abstract math concepts can have tangible impacts on our lives and the world around us.

The key takeaway here is that mastering the basics is essential for future success in mathematics and beyond. Systems of equations are a fundamental topic, and a solid understanding of them will serve you well as you tackle more advanced concepts in algebra, calculus, and other areas of mathematics. The skills you've developed in solving this system – problem-solving, critical thinking, attention to detail – are transferable skills that will benefit you in any field you choose to pursue.

Remember, mathematics is not just about memorizing formulas and procedures; it's about developing a way of thinking. It's about breaking down complex problems into smaller, manageable steps, applying logical reasoning, and arriving at solutions. It's also about appreciating the beauty and elegance of mathematical concepts and their connections to the world around us.

So, keep practicing, keep exploring, and keep asking questions. Don't be afraid to make mistakes – they're a natural part of the learning process. The more you engage with mathematics, the more confident and proficient you'll become. And who knows, maybe one day you'll be using systems of equations to solve a groundbreaking problem or invent the next great technology. The possibilities are endless!