Solving Systems Of Equations A Detailed Solution Of {3x - 2y = 12, X + 2y = 12}

Hey guys! Today, we're diving deep into the fascinating world of solving systems of equations. Specifically, we're going to tackle the system ${\left\{\begin{array}{c}3 x-2 y=12 \\ x+2 y=12\end{array}\right.}$. Don't worry if that looks intimidating—by the end of this article, you'll be a pro at solving these types of problems. We'll break it down step-by-step, making sure everyone, from beginners to seasoned math enthusiasts, can follow along. So, grab your pencils, and let's get started!

Understanding Systems of Equations

Before we jump into the nitty-gritty, let's make sure we're all on the same page about what a system of equations actually is. In the simplest terms, a system of equations is a set of two or more equations that share the same variables. Our goal? To find the values of those variables that satisfy all equations in the system simultaneously. Think of it as a puzzle where all the pieces need to fit together perfectly. In our specific system, we have two equations:

1. ${3x - 2y = 12}$
2. ${x + 2y = 12}$

We need to find the values of ${x}$ and ${y}$ that make both of these equations true. There are several methods we can use to solve systems of equations, each with its own strengths and weaknesses. We'll be focusing on the elimination method here because it's particularly efficient for this system. The beauty of the elimination method lies in its ability to strategically manipulate equations to cancel out one variable, making it easier to solve for the other. Before we dive into the solution, it's important to understand why solving systems of equations is crucial. These systems pop up in various real-world scenarios, from calculating costs and profits in business to modeling physical phenomena in science and engineering. Mastering the techniques to solve them opens doors to a wide array of applications. Imagine you're trying to determine the break-even point for a new product, or you're analyzing the trajectory of a projectile. Systems of equations are the unsung heroes behind these calculations. So, let's embark on this journey together, and unlock the power of systems of equations!

The Elimination Method: A Step-by-Step Approach

Alright, let's get down to the heart of the matter: using the elimination method to solve our system. The elimination method, guys, is like a mathematical magic trick where we strategically add or subtract equations to get rid of one variable. When looking at our system:

1. ${3x - 2y = 12}$
2. ${x + 2y = 12}$

You might notice something pretty cool. The ${y}$ terms in the two equations have opposite signs (-2y and +2y). This is fantastic news because it means we're already set up perfectly for elimination! The first step in the elimination method is to make sure that the coefficients of one of the variables are opposites. In our case, the coefficients of ${y}$ are already -2 and +2, so we can skip that step and move straight to the next one. Next, we simply add the two equations together. When we do this, the ${y}$ terms will cancel each other out, leaving us with an equation in just one variable (${x}$). Let's see how it works:

${(3x - 2y) + (x + 2y) = 12 + 12}$

Simplifying the left side, the -2y and +2y terms vanish, and we're left with:

${4x = 24}$

Now, we have a simple equation to solve for ${x}$. Divide both sides by 4:

${x = 6}$

Boom! We've found the value of ${x}$. But we're not done yet. We still need to find the value of ${y}$. Once we've solved for one variable, we can substitute its value back into any of the original equations to solve for the other variable. This is the beauty of the elimination method – it simplifies the problem into manageable steps. Choosing which equation to substitute into is up to you, but it's often easiest to pick the equation that looks simpler or has smaller coefficients. In this case, the second equation, ${x + 2y = 12}$, seems like a good choice. So, let's substitute ${x = 6}$ into that equation and solve for ${y}$.

Solving for the Remaining Variable and Verifying the Solution

Now that we've successfully found the value of ${x}$, which is 6, the next step is to substitute this value into one of our original equations to solve for ${y}$. As we discussed earlier, the second equation, ${x + 2y = 12}$, looks like the simplest option. So, let's plug in ${x = 6}$:

${6 + 2y = 12}$

To isolate ${y}$, we first subtract 6 from both sides of the equation:

${2y = 6}$

Then, we divide both sides by 2:

${y = 3}$

Awesome! We've found that ${y = 3}$. So, our solution to the system of equations is ${x = 6}$ and ${y = 3}$. We can write this as an ordered pair: ${(6, 3)}$. But before we celebrate our victory, there's one crucial step we need to take: verification. It's always a good idea to check our solution to make sure it actually works. To do this, we substitute both values, ${x = 6}$ and ${y = 3}$, back into both of the original equations. If both equations hold true, then we know we've found the correct solution. Let's start with the first equation:

${3x - 2y = 12}$

Substitute ${x = 6}$ and ${y = 3}$:

${3(6) - 2(3) = 18 - 6 = 12}$

Great! The first equation holds true. Now, let's check the second equation:

${x + 2y = 12}$

Substitute ${x = 6}$ and ${y = 3}$:

${6 + 2(3) = 6 + 6 = 12}$

Fantastic! The second equation also holds true. Since our solution satisfies both equations, we can confidently say that ${(6, 3)}$ is the correct solution to the system. Always remember, guys, verification is a key step in problem-solving. It's like having a safety net that catches any potential errors. So, never skip it!

Alternative Methods and When to Use Them

While we've focused on the elimination method here, it's important to know that there are other ways to solve systems of equations. Each method has its strengths and is better suited for certain types of systems. Let's briefly touch on two other common methods: substitution and graphing. The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This method is particularly useful when one of the equations is already solved for a variable or can be easily rearranged. For example, if we had the system:

${\left\{\begin{array}{c}y = 2x + 1 \\ 3x + y = 10\end{array}\right.}$

The first equation is already solved for ${y}$, so substitution would be a natural choice. We would substitute ${2x + 1}$ for ${y}$ in the second equation and solve for ${x}$. Then, we'd substitute the value of ${x}$ back into either equation to find ${y}$. The graphing method, on the other hand, involves plotting the equations on a coordinate plane. The solution to the system is the point where the lines intersect. This method is great for visualizing the system and understanding the relationship between the equations. However, it's not always the most accurate method, especially if the intersection point has non-integer coordinates. Graphing is most effective when you need a quick estimate of the solution or when you want to understand the graphical representation of the system. So, when should you use each method? The elimination method shines when the coefficients of one variable are opposites or can be easily made opposites by multiplying one or both equations by a constant, as we saw in our example. The substitution method is ideal when one equation is already solved for a variable or can be easily solved. And the graphing method is best for visualization and estimation. Ultimately, the best method depends on the specific system you're dealing with and your personal preference. The more you practice with different methods, the better you'll become at choosing the most efficient one for each problem. So, keep exploring, keep practicing, and keep honing your problem-solving skills!

Real-World Applications of Systems of Equations

Okay, guys, so we've mastered the techniques for solving systems of equations, but you might be wondering, "Where does this stuff actually show up in the real world?" Well, the truth is, systems of equations are all around us! They're used in a surprisingly wide range of fields, from business and economics to science and engineering. Let's explore a few examples to see how these mathematical tools can help us solve real-world problems. One common application is in business and economics. Imagine you're running a small business that sells two products, let's say Product A and Product B. You know the cost of producing each product and the selling price. You also have a budget for production and a target revenue goal. You can set up a system of equations to determine how many units of each product you need to sell to meet your goals. For example, one equation might represent the total cost of production, and the other equation might represent the total revenue. By solving this system, you can find the optimal production quantities for each product. Systems of equations also play a crucial role in science and engineering. In physics, they can be used to analyze the motion of objects, the flow of fluids, and the behavior of electrical circuits. For instance, you might use a system of equations to determine the forces acting on an object or the currents flowing through different parts of a circuit. In chemistry, systems of equations can be used to balance chemical equations and to calculate the amounts of reactants and products in a chemical reaction. Imagine you're designing a bridge or a building. Systems of equations are essential for calculating the stresses and strains on different parts of the structure, ensuring its stability and safety. In computer science, systems of equations are used in computer graphics, game development, and artificial intelligence. For example, they can be used to model the movement of characters in a video game or to train machine learning algorithms. Systems of equations even pop up in everyday life. Think about planning a trip. You might have a budget for transportation and accommodation, and you want to find the best combination of flights and hotels that fits your budget. You can set up a system of equations to represent your budget constraints and find the optimal travel plan. These are just a few examples, guys, but they illustrate the versatility and power of systems of equations. They're not just abstract mathematical concepts; they're tools that can help us understand and solve real-world problems. So, the next time you encounter a situation that involves multiple variables and constraints, remember the power of systems of equations!

Conclusion and Further Practice

Alright, guys, we've reached the end of our journey into the world of solving systems of equations! We've covered the elimination method in detail, learned how to verify our solutions, and even explored other methods and real-world applications. You've armed yourselves with a powerful tool for tackling mathematical challenges. Remember, the key to mastering any mathematical concept is practice. The more you practice, the more comfortable and confident you'll become. So, don't stop here! Seek out more systems of equations to solve, and try different methods to see what works best for you. There are tons of resources available online and in textbooks. Look for practice problems, quizzes, and even interactive games that can help you hone your skills. Challenge yourself with increasingly complex systems, and don't be afraid to make mistakes. Mistakes are a natural part of the learning process. The important thing is to learn from them and keep moving forward. If you get stuck on a problem, don't hesitate to ask for help. Talk to your teachers, classmates, or online communities. There are plenty of people who are willing to share their knowledge and expertise. Also, remember that understanding the underlying concepts is just as important as memorizing the steps. Try to think about why each step works, rather than just blindly following a procedure. This will help you develop a deeper understanding of systems of equations and how they relate to other mathematical concepts. Solving systems of equations is a valuable skill that will serve you well in many areas of life, from academics to your future career. So, embrace the challenge, enjoy the process, and keep practicing! And who knows, maybe one day you'll be using systems of equations to solve some of the world's most pressing problems. Keep exploring the fascinating world of mathematics, guys, and never stop learning! You've got this!