Solving Systems Of Equations Step-by-Step 2x-y=24 And X+y=6
Hey guys! 👋 Ever get stuck trying to solve those tricky systems of equations? You know, the ones with two equations and two unknowns? Don't worry, we've all been there! Today, we're going to break down a common problem: solving the system of equations 2x - y = 24 and x + y = 6. We'll go through it step-by-step, so you can ace your next math test or just impress your friends with your awesome algebra skills. 😎
Understanding Systems of Equations
First, let's get a grip on what we're actually dealing with. A system of equations is simply a set of two or more equations that share the same variables. In our case, we have two equations, and both use the variables 'x' and 'y'. The solution to a system of equations is a set of values for the variables that makes all the equations true at the same time. Think of it like finding the perfect 'x' and 'y' that fit both equations like a glove! There are a few common methods to tackle these problems, but we'll focus on one of the most popular and straightforward: the elimination method.
The Power of Elimination: Making Variables Disappear
The elimination method, also sometimes called the addition method, is a neat trick that involves manipulating the equations in your system so that when you add them together, one of the variables cancels out. Poof! Gone! This leaves you with a single equation with just one variable, which is super easy to solve. Then, once you know the value of that variable, you can plug it back into one of the original equations to find the value of the other variable. It's like a mathematical magic trick!✨
The beauty of the elimination method lies in its strategic approach. We look for opportunities to create opposite coefficients for one of the variables. Coefficients are just the numbers that multiply the variables (like the '2' in '2x'). If we can get the coefficients of either 'x' or 'y' to be opposites (like 3 and -3), then when we add the equations, those terms will disappear. Let's see how this works in our example.
Solving 2x - y = 24 and x + y = 6 using Elimination
Okay, let's dive into our specific problem:
- Equation 1: 2x - y = 24
- Equation 2: x + y = 6
Take a close look at those equations. Notice anything special? 👀 The 'y' terms have coefficients of -1 and +1. Bingo! They're already opposites! This means we can jump right into the addition step. No extra manipulation needed for this one, how cool is that?
Step 1: Adding the Equations
This is the fun part! We're going to add the left-hand sides of the equations together and set that equal to the sum of the right-hand sides. It looks like this:
(2x - y) + (x + y) = 24 + 6
Now, let's simplify. Combine the 'x' terms: 2x + x = 3x. And here's where the magic happens: the 'y' terms, -y and +y, cancel each other out! 🥳 This is why we call it the elimination method. On the right side, 24 + 6 = 30. So, our equation simplifies to:
3x = 30
See? One simple equation with one variable. We're making progress!
Step 2: Solving for x
Now we have a simple equation: 3x = 30. To isolate 'x', we need to get rid of that '3' that's multiplying it. We do this by dividing both sides of the equation by 3:
(3x) / 3 = 30 / 3
This simplifies to:
x = 10
We've found it! 🎉 The value of 'x' that satisfies both equations is 10. Half the battle is won!
Step 3: Solving for y
Now that we know x = 10, we can plug this value back into either of our original equations to solve for 'y'. It doesn't matter which equation you choose; you'll get the same answer either way. Let's use Equation 2 (x + y = 6) because it looks a little simpler:
10 + y = 6
To isolate 'y', we need to subtract 10 from both sides:
10 + y - 10 = 6 - 10
This simplifies to:
y = -4
And there you have it! We've found the value of 'y': it's -4.
Step 4: The Solution and a Quick Check
We've solved the system of equations! The solution is x = 10 and y = -4. We often write this as an ordered pair: (10, -4). This means that the point (10, -4) is the intersection of the two lines represented by our equations. Cool, huh? 😎
But before we celebrate too much, let's make sure we didn't make any mistakes. A quick check is always a good idea. We'll plug our values for 'x' and 'y' back into both original equations to see if they hold true.
- Equation 1: 2x - y = 24
- 2(10) - (-4) = 20 + 4 = 24 ✅ (It works!)
- Equation 2: x + y = 6
- 10 + (-4) = 6 ✅ (It works again!)
Both equations are satisfied, so we can be confident that our solution is correct. High five! 🙌
When Elimination Needs a Little Help: Multiplication Magic
Okay, so our first example was pretty straightforward. The 'y' terms were already opposites, so we could jump right into adding the equations. But what if the coefficients aren't so cooperative? What if we don't have any terms that are opposites right away? That's where a little multiplication magic comes in.
Sometimes, to use the elimination method, you'll need to multiply one or both equations by a constant. This doesn't change the solution to the system, but it does change the coefficients, allowing you to create those crucial opposite terms. Let's look at a hypothetical example to see how this works.
Imagine we had these equations:
- 3x + 2y = 7
- x - y = -1
Notice that neither the 'x' coefficients (3 and 1) nor the 'y' coefficients (2 and -1) are opposites. We can't add these equations directly and eliminate a variable. But we can make it happen! 🧙
Look at the 'y' coefficients: 2 and -1. If we could turn that -1 into a -2, then we'd have opposites! How do we do that? We multiply the entire second equation by 2:
2 * (x - y) = 2 * (-1)
This gives us a new equation:
2x - 2y = -2
Now we have a new system of equations:
- 3x + 2y = 7
- 2x - 2y = -2
Ta-da! ✨ The 'y' coefficients are now opposites (+2 and -2). We can now add these equations together and eliminate 'y', just like we did in our first example. We'd get:
5x = 5
And we could solve for 'x' (x = 1) and then plug that value back in to find 'y'.
Key Takeaways for Mastering Elimination
Alright, guys, we've covered a lot! Let's recap the key steps for solving systems of equations using the elimination method:
- Line up the variables: Make sure your equations are written in the same form (usually with the 'x' and 'y' terms on one side and the constant on the other). This helps you keep things organized.
- Look for opposites (or create them!): Check if any of your variables already have opposite coefficients. If not, think about multiplying one or both equations by a constant to create those opposites.
- Add the equations: Add the left-hand sides together and set that equal to the sum of the right-hand sides. The variable with opposite coefficients should disappear!
- Solve for the remaining variable: You'll now have a simple equation with just one variable. Solve for that variable.
- Substitute back in: Plug the value you just found back into one of the original equations to solve for the other variable.
- Write the solution as an ordered pair: Express your solution as (x, y).
- Check your work: The most important step. Plug your values for 'x' and 'y' back into both original equations to make sure they hold true.
Why is Solving Systems of Equations Important?
You might be thinking, "Okay, this is cool, but why do I need to know this?" Well, solving systems of equations isn't just some abstract math concept. It has real-world applications in fields like:
- Science and Engineering: Many scientific and engineering problems involve multiple variables and relationships that can be modeled as systems of equations.
- Economics: Economists use systems of equations to model supply and demand, market equilibrium, and other economic phenomena.
- Computer Graphics: Systems of equations are used to perform transformations and calculations in computer graphics.
- Everyday Life: Even in everyday situations, you might encounter problems that can be solved using systems of equations. For example, figuring out the cost of two different items given their combined price and the difference in their prices.
So, the skills you learn in solving systems of equations are valuable and can be applied in many different contexts.
Practice Makes Perfect! 💪
The best way to master the elimination method (or any math skill, really) is to practice! Grab some practice problems from your textbook, online resources, or even make up your own. The more you work through different types of problems, the more comfortable and confident you'll become. Don't be afraid to make mistakes – that's how we learn! And if you get stuck, remember the steps we've discussed, and don't hesitate to ask for help from your teacher, a tutor, or a friend.
Keep practicing, and you'll be a system-of-equations-solving pro in no time! 😉 You've got this!
