Solving Systems Of Equations Elimination Method Example 4x-5y=2 And 3x-5y=-1
Hey guys! Today, we're diving into a super important topic in math: solving systems of equations. Specifically, we're going to break down a problem that looks like this: 4x - 5y = 2 and 3x - 5y = -1. Don't worry if it seems intimidating at first; we'll take it step by step and you'll see it's actually pretty straightforward. We'll be focusing on a method called the elimination method, which is a powerful tool for tackling these kinds of problems. So, buckle up and let's get started!
Understanding Systems of Equations
Before we jump into the solution, let's make sure we're all on the same page about what a system of equations actually is. Think of it as a set of two or more equations that share the same variables. In our case, we have two equations, and the variables are 'x' and 'y'. The goal here is to find values for 'x' and 'y' that satisfy both equations simultaneously. This means that when you plug those values into both equations, they both have to be true. There are several ways to solve these systems, but today we're zeroing in on the elimination method. This method is particularly useful when you notice that the coefficients of one of the variables are the same or can easily be made the same. This allows us to eliminate one variable by adding or subtracting the equations, making it easier to solve for the remaining variable. Understanding this foundational concept is key to mastering the elimination method, so make sure you've got this down before moving on.
When dealing with systems of linear equations, we're essentially looking for the point where two or more lines intersect on a graph. Each equation represents a line, and the solution to the system is the coordinates (x, y) of the point where those lines cross each other. There are three possible scenarios: the lines intersect at one point (one unique solution), the lines are parallel and never intersect (no solution), or the lines are the same (infinitely many solutions). The elimination method helps us find that intersection point, if it exists. By strategically manipulating the equations, we can eliminate one variable and solve for the other. This gives us one coordinate of the intersection point, and we can then substitute that value back into one of the original equations to find the other coordinate. This visual and geometric interpretation can be incredibly helpful in understanding what we're actually doing when we solve these equations algebraically. It's not just about manipulating numbers; it's about finding a point that lies on both lines.
Furthermore, the concept of solving systems of equations extends far beyond the classroom. It's a fundamental tool used in various fields like engineering, economics, computer science, and many more. For instance, engineers might use systems of equations to model the forces acting on a structure, while economists might use them to analyze supply and demand curves. Computer scientists use them in algorithms for optimization and machine learning. The ability to solve these systems efficiently and accurately is a valuable skill in many professional contexts. So, the time and effort you invest in mastering these techniques will definitely pay off in the long run. It's not just about getting the right answer on a test; it's about developing a powerful problem-solving tool that you can apply to real-world situations. The more comfortable you become with these methods, the better equipped you'll be to tackle complex challenges in various fields.
The Elimination Method: A Step-by-Step Guide
Okay, let's dive into the elimination method itself. This method is all about strategically adding or subtracting equations to get rid of one variable. In our case, we have these equations:
- 4x - 5y = 2
- 3x - 5y = -1
Notice anything interesting? Look at the 'y' terms. Both equations have '-5y'. This is perfect for elimination! The goal is to make the coefficients of either 'x' or 'y' the same (or opposites) so that when we add or subtract the equations, one variable cancels out. In this case, the coefficients of 'y' are already the same, which saves us a step. This is a key advantage of the elimination method: it can be very efficient when the equations are set up in a way that makes elimination easy. The ability to spot these opportunities quickly comes with practice, so don't be discouraged if it doesn't click right away. The more you work with these systems, the better you'll become at recognizing patterns and choosing the most efficient approach. Remember, the ultimate goal is to simplify the problem and make it easier to solve, and the elimination method is a powerful tool for achieving that.
To eliminate 'y', we can subtract the second equation from the first equation. Why subtract? Because -5y - (-5y) = 0. This is the magic of the elimination method! When we subtract the equations, the 'y' terms disappear, leaving us with an equation that only involves 'x'. This is a crucial step in solving the system because it reduces the problem from two variables to one, which is much easier to handle. It's like taking a complex puzzle and breaking it down into smaller, more manageable pieces. The key is to carefully perform the subtraction, making sure to subtract each term in the second equation from the corresponding term in the first equation. This will give you a new equation in terms of 'x', which you can then solve using basic algebraic techniques. The ability to perform this subtraction accurately is essential for the success of the elimination method, so pay close attention to the signs and make sure you're subtracting the entire equation.
Subtracting the equations, we get:
(4x - 5y) - (3x - 5y) = 2 - (-1)
Let's simplify this. Remember to distribute the negative sign when subtracting the second equation:
4x - 5y - 3x + 5y = 2 + 1
Notice how the '-5y' and '+5y' cancel each other out? That's exactly what we wanted! Now we have:
x = 3
Wow, we've already found the value of 'x'! This is a major step forward. The elimination method has allowed us to isolate 'x' and determine its value directly. Now that we know 'x', we can move on to the next phase of the problem: finding the value of 'y'. This is where the power of substitution comes in. We can take the value of 'x' that we just found and plug it back into one of the original equations to solve for 'y'. This is a common technique in solving systems of equations, and it highlights the interconnectedness of the variables. Knowing one variable allows us to unlock the value of the other. The key is to choose the equation that looks easiest to work with, as this will minimize the chance of making errors in the substitution and simplification process. So, let's choose an equation and see what we get for 'y'.
Solving for 'y'
Now that we know x = 3, we can substitute this value into either of the original equations to solve for 'y'. Let's use the first equation, 4x - 5y = 2. It doesn't really matter which equation you choose; you'll get the same answer for 'y' either way. However, sometimes one equation might look simpler than the other, which can make the substitution and simplification process a bit easier. The key is to choose the equation that you feel most comfortable working with. This is where your problem-solving intuition comes into play. With practice, you'll develop a sense for which equations are likely to lead to the easiest calculations. So, don't be afraid to experiment and try different approaches. The more you practice, the better you'll become at making these kinds of decisions.
Substituting x = 3 into 4x - 5y = 2, we get:
4(3) - 5y = 2
12 - 5y = 2
Now we have a simple equation with just one variable, 'y'. To solve for 'y', we need to isolate it on one side of the equation. This involves performing a series of algebraic operations, such as adding or subtracting terms from both sides. The goal is to undo the operations that are currently being applied to 'y', one step at a time. Think of it like peeling back the layers of an onion. Each step brings you closer to the core value of 'y'. It's important to remember that whatever operation you perform on one side of the equation, you must also perform on the other side to maintain the balance. This is a fundamental principle of algebra, and it's crucial for solving equations correctly. So, let's see how we can isolate 'y' in this particular equation.
Subtract 12 from both sides:
-5y = 2 - 12
-5y = -10
Finally, divide both sides by -5:
y = -10 / -5
y = 2
Great! We've found the value of 'y'. So, the solution to our system of equations is x = 3 and y = 2. This means that the point (3, 2) is the intersection point of the two lines represented by our original equations. This is a crucial piece of information because it tells us that this is the only pair of values for 'x' and 'y' that will satisfy both equations simultaneously. We've successfully navigated the elimination method and arrived at the solution. But, before we celebrate, there's one important step we need to take to ensure that our answer is correct.
Checking Our Solution
It's always a good idea to check your solution to make sure you didn't make any mistakes along the way. This is a crucial step in the problem-solving process, and it can save you from submitting an incorrect answer. Checking your solution is like having a built-in error detector. It allows you to verify that the values you've found for 'x' and 'y' actually satisfy both of the original equations. This gives you confidence that you've solved the system correctly. The process is simple: just plug the values of 'x' and 'y' back into the original equations and see if they hold true. If both equations are satisfied, then you've got the right solution. If not, then you know there's a mistake somewhere, and you need to go back and review your steps.
To check our solution (x = 3, y = 2), we'll plug these values into both original equations:
Equation 1: 4x - 5y = 2
4(3) - 5(2) = 12 - 10 = 2 (This is correct!)
Equation 2: 3x - 5y = -1
3(3) - 5(2) = 9 - 10 = -1 (This is also correct!)
Since our solution satisfies both equations, we can be confident that it's correct. Hooray! We've successfully solved the system of equations using the elimination method. This is a fantastic feeling, knowing that you've tackled a challenging problem and come up with the right answer. But the journey doesn't end here. The more you practice solving these kinds of problems, the more comfortable and confident you'll become. So, keep practicing and exploring different types of systems of equations. You'll be amazed at how quickly you improve and how much more comfortable you become with these techniques.
Conclusion
So, there you have it! We've walked through the process of solving the system of equations 4x - 5y = 2 and 3x - 5y = -1 using the elimination method. Remember, the key is to identify opportunities to eliminate a variable by adding or subtracting equations. We found that x = 3 and y = 2 is the solution that satisfies both equations. This method is a powerful tool in your math arsenal, and with practice, you'll become a pro at solving these types of problems. Keep up the great work, guys, and happy problem-solving!
