Solving -4x + 3y = 49 And -x - 2y = 4 A Step By Step Guide

Hey guys! Let's dive into solving a system of linear equations. Today, we're tackling the equations -4x + 3y = 49 and -x - 2y = 4. These types of problems pop up everywhere in math and science, so understanding how to solve them is super important. We'll break it down step-by-step, making sure everyone can follow along. We will explore different methods to solve this system, ensuring a solid understanding of the underlying concepts.

Understanding Systems of Equations

So, what exactly is a system of equations? Well, it’s basically a set of two or more equations that share the same variables. Our goal is to find the values for these variables that make all the equations true at the same time. Think of it like a puzzle where all the pieces need to fit together perfectly. In our case, we have two equations with two variables, x and y. The solutions we're looking for are the specific values of x and y that satisfy both equations simultaneously. There are several methods to approach these systems, such as substitution, elimination, and graphing. Each method has its own strengths, and choosing the right one can make the solving process much smoother. For instance, if one equation is already solved for one variable, the substitution method might be the most efficient. On the other hand, if the coefficients of one variable are opposites or can easily be made opposites, the elimination method could be the better choice. Before we jump into solving, it's worth mentioning that a system of equations can have one solution, infinitely many solutions, or no solution at all. If the lines intersect at one point, we have a unique solution. If the lines are the same, we have infinitely many solutions. And if the lines are parallel, we have no solution. Understanding these possibilities helps us interpret our results once we’ve crunched the numbers. So, let’s get started and see how we can crack this particular system of equations!

Method 1: Substitution

The substitution method is a handy way to solve systems of equations. The basic idea is to solve one equation for one variable and then substitute that expression into the other equation. This leaves us with a single equation with just one variable, which we can then easily solve. Let's see how it works with our equations, -4x + 3y = 49 and -x - 2y = 4. First, we need to pick one equation and solve it for one of the variables. Looking at the second equation, -x - 2y = 4, it seems easiest to solve for x because the coefficient of x is already -1. Adding 2y to both sides gives us -x = 2y + 4. Then, multiplying both sides by -1, we get x = -2y - 4. Great! We've now expressed x in terms of y. Next up, we'll substitute this expression for x into the first equation, -4x + 3y = 49. Replacing x with (-2y - 4) gives us -4(-2y - 4) + 3y = 49. Now we need to simplify and solve for y. Distributing the -4, we get 8y + 16 + 3y = 49. Combining like terms, we have 11y + 16 = 49. Subtracting 16 from both sides gives us 11y = 33, and finally, dividing by 11, we find that y = 3. Awesome! We've found the value of y. Now that we know y, we can plug it back into either of our original equations (or the expression we found for x) to solve for x. Let's use the expression x = -2y - 4 since we already have it. Substituting y = 3, we get x = -2(3) - 4, which simplifies to x = -6 - 4, so x = -10. So, our solution is x = -10 and y = 3. We can write this as the ordered pair (-10, 3). To make sure we're right, we can plug these values back into both original equations to check. For the first equation, -4x + 3y = 49, we have -4(-10) + 3(3) = 40 + 9 = 49, which is correct. For the second equation, -x - 2y = 4, we have -(-10) - 2(3) = 10 - 6 = 4, which is also correct. So, we've successfully solved the system using substitution! We found that x = -10 and y = 3 is the solution that works for both equations. Let's explore another method to solve this system, just to see how different approaches can lead to the same answer.

Method 2: Elimination

Another fantastic method for tackling systems of equations is the elimination method, sometimes called the addition method. The core idea here is to manipulate the equations so that when we add them together, one of the variables cancels out. This leaves us with a single equation with one variable, just like in the substitution method. Let’s apply this to our system: -4x + 3y = 49 and -x - 2y = 4. Looking at our equations, we need to think about how to make either the x coefficients or the y coefficients opposites. It looks like it would be easier to eliminate x in this case. We have -4x in the first equation and -x in the second. If we multiply the second equation by -4, we'll get 4x, which is the opposite of -4x. So, let's multiply the entire second equation, -x - 2y = 4, by -4. This gives us 4x + 8y = -16. Now we have our modified system:

• -4x + 3y = 49
• 4x + 8y = -16

Now, we can add the two equations together. The -4x and 4x terms cancel out, leaving us with 3y + 8y = 49 + (-16). Combining like terms, we get 11y = 33. Dividing both sides by 11, we find y = 3. Excellent! We've got the value of y, which matches what we found using substitution. Next, we need to find x. We can plug y = 3 back into either of our original equations. Let's use the second original equation, -x - 2y = 4. Substituting y = 3, we have -x - 2(3) = 4, which simplifies to -x - 6 = 4. Adding 6 to both sides gives us -x = 10, and multiplying by -1 gives us x = -10. Just like before, we found x = -10 and y = 3. Our solution, written as an ordered pair, is (-10, 3). To double-check, let’s plug these values into both original equations. For the first equation, -4x + 3y = 49, we have -4(-10) + 3(3) = 40 + 9 = 49, which is correct. For the second equation, -x - 2y = 4, we have -(-10) - 2(3) = 10 - 6 = 4, which is also correct. So, the elimination method confirms our solution. We've now seen two different methods, substitution and elimination, both leading to the same answer. This highlights the power of having multiple tools in our math toolkit. Let’s summarize our findings and wrap things up.

Solution and Verification

Alright, after working through both the substitution and elimination methods, we've arrived at the same solution: x = -10 and y = 3. This can be written as the ordered pair (-10, 3). But before we declare victory, it's always a good idea to verify our solution. This means plugging our values for x and y back into the original equations to make sure they hold true. Let's start with the first equation: -4x + 3y = 49. Substituting x = -10 and y = 3, we get -4(-10) + 3(3) = 40 + 9 = 49. Bingo! It checks out. Now, let's move on to the second equation: -x - 2y = 4. Plugging in our values, we have -(-10) - 2(3) = 10 - 6 = 4. Another success! Since our solution satisfies both equations, we can confidently say that (-10, 3) is indeed the solution to the system. This verification step is super important because it helps us catch any mistakes we might have made along the way. It's like the final piece of the puzzle that confirms everything fits perfectly. Sometimes, errors can creep in during the calculations, so taking the time to verify can save us from incorrect answers. Moreover, verifying our solution reinforces our understanding of what it means to solve a system of equations. We're not just finding numbers; we're finding the specific values that make both equations true simultaneously. This gives us a deeper appreciation for the elegance and precision of mathematics. So, remember guys, always verify your solutions! It's a simple step that can make a big difference in your accuracy and understanding. Now that we've confidently solved and verified our system, let's recap the key steps we took and discuss some general strategies for tackling similar problems.

Tips and Tricks for Solving Systems of Equations

Solving systems of equations might seem daunting at first, but with practice and a few handy tips and tricks, it becomes much easier. We've already seen two powerful methods – substitution and elimination – and knowing when to use each one can make a big difference. So, let's break down some strategies to help you conquer these problems. First off, when you're faced with a system of equations, take a moment to assess the equations. Look for opportunities to simplify or rearrange them. If one of the equations is already solved for a variable, or if it's easy to isolate a variable, the substitution method might be your best bet. On the other hand, if you notice that the coefficients of one variable are opposites or can easily be made opposites by multiplying an equation, the elimination method could be more efficient. Another useful trick is to be organized in your work. Write down each step clearly, and double-check your calculations as you go. This helps prevent errors and makes it easier to track your progress. It's also a good idea to use consistent notation, like always writing your variables in the same order. Remember the importance of the verification step. Always plug your solution back into the original equations to make sure it works. This not only confirms your answer but also helps you catch any mistakes. If your solution doesn't check out, go back and review your steps to see where you might have gone wrong. Don't be afraid to try a different method if one approach isn't working. Sometimes, one method is significantly easier than the other, and experimenting can save you time and effort. If you get stuck, try graphing the equations. The point of intersection represents the solution, and this can give you a visual understanding of the problem. Moreover, practice makes perfect! The more systems of equations you solve, the more comfortable you'll become with the different methods and strategies. Look for practice problems in your textbook or online, and don't hesitate to ask for help from your teacher or classmates if you need it. Finally, remember that solving systems of equations is a valuable skill that extends beyond the classroom. It's used in many real-world applications, from engineering and economics to computer science and beyond. So, the effort you put into mastering these techniques will pay off in many ways. By following these tips and tricks, you'll be well-equipped to tackle any system of equations that comes your way. Let's recap the main points we've covered in this discussion.

Conclusion

In this comprehensive guide, we've thoroughly explored how to solve the system of equations -4x + 3y = 49 and -x - 2y = 4. We kicked things off by understanding what a system of equations is and why it's important in mathematics and various real-world applications. Then, we dived into two powerful methods for solving such systems: substitution and elimination. With the substitution method, we solved one equation for one variable and plugged that expression into the other equation, reducing the problem to a single equation with one variable. This method is particularly useful when one of the equations is already solved for a variable or when it's easy to isolate one. Next, we tackled the system using the elimination method, where we manipulated the equations so that adding them together would eliminate one of the variables. This method is especially effective when the coefficients of one variable are opposites or can be easily made opposites. We saw how multiplying one or both equations by a constant can set up the elimination beautifully. Throughout the process, we emphasized the importance of organization and clear steps. Writing down each step carefully and double-checking calculations can prevent errors and make the process much smoother. We also highlighted the crucial step of verification. Plugging our solution back into the original equations ensures that our answer is correct and helps us catch any mistakes we might have made along the way. Verification not only confirms our solution but also deepens our understanding of what it means for a solution to satisfy a system of equations. Furthermore, we shared some valuable tips and tricks for solving systems of equations. These include assessing the equations to choose the most efficient method, being mindful of organization, trying different approaches if needed, and practicing regularly to build confidence and fluency. Remember, there's no one-size-fits-all approach, and sometimes a combination of methods might be the best strategy. By mastering these techniques, you'll be well-prepared to tackle a wide range of problems involving systems of equations. This skill is not only essential in mathematics but also in various fields that rely on mathematical modeling and problem-solving. So, keep practicing, stay curious, and remember that every challenge is an opportunity to learn and grow. We hope this guide has been helpful in your journey to mastering systems of equations. Happy solving, guys!