Solving X+2y=10 And 3x+4y=24 A Step-by-Step Guide Using Equalization Method
Hey guys! Today, we're diving into the fascinating world of algebra to tackle a system of equations. Specifically, we'll be solving the system:
- x + 2y = 10
- 3x + 4y = 24
We're going to use the equalization method, which is a super handy technique for finding the values of our unknowns, x and y. So, buckle up and let's get started!
Understanding the Equalization Method
Before we jump into the solution, let's quickly chat about what the equalization method actually is. Think of it like this: we're trying to isolate the same variable in both equations. Once we have that variable by itself in each equation, we can then set the two expressions equal to each other. This creates a new equation with only one variable, which is something we can easily solve! In essence, the equalization method is a smart way to eliminate one variable, making it easier to solve for the other. It's a powerful tool in your algebraic arsenal, and once you get the hang of it, you'll be solving systems of equations like a pro. We can then substitute the value we found back into one of the original equations to solve for the remaining variable. This might sound a bit abstract right now, but trust me, it'll all make sense once we walk through the steps. Remember, the key is to manipulate the equations strategically so that we can isolate the same variable in both. This might involve multiplying one or both equations by a constant, or adding or subtracting terms from both sides. The goal is always to get that variable by itself, ready to be equated with the expression from the other equation. And don't worry if it seems tricky at first – practice makes perfect! The more you work with systems of equations and the equalization method, the more comfortable and confident you'll become. Soon, you'll be able to spot the best way to approach a problem and solve it quickly and efficiently. So, let's keep this in mind as we move forward and tackle our specific system of equations. We'll break it down step by step, and you'll see just how effective this method can be.
Step-by-Step Solution
Step 1: Isolate a Variable in Both Equations
The first crucial step in the equalization method is to isolate the same variable in both equations. This is where we get to play around with the equations and use our algebraic skills to get the variable we want all by itself on one side of the equals sign. Looking at our system:
- x + 2y = 10
- 3x + 4y = 24
It seems like isolating x might be the easier route here. In the first equation, x is already pretty close to being alone. In the second equation, we'll need to do a little more work, but it's definitely manageable. So, let's start with the first equation, x + 2y = 10. To isolate x, we simply need to subtract 2y from both sides. This gives us:
x = 10 - 2y
Great! We've got x isolated in the first equation. Now, let's move on to the second equation, 3x + 4y = 24. Here, we also want to get x by itself. The first thing we can do is subtract 4y from both sides:
3x = 24 - 4y
Now, we have 3x isolated, but we want just x. To get rid of the 3, we divide both sides of the equation by 3:
x = (24 - 4y) / 3
Fantastic! We've now isolated x in both equations. We have:
- x = 10 - 2y
- x = (24 - 4y) / 3
This is exactly what we need for the next step. Remember, the key here is to be methodical and careful with our algebraic manipulations. Make sure you're performing the same operation on both sides of the equation to maintain balance. And don't be afraid to take your time and double-check your work. A small mistake in this step can throw off the whole solution. But with a little practice, you'll become a master at isolating variables and setting up your equations for the next step in the equalization method.
Step 2: Set the Expressions Equal to Each Other
Alright, guys, we've reached a super important step in the equalization method! We've successfully isolated x in both of our equations, and now we're ready to put those expressions together. Remember, we have:
- x = 10 - 2y
- x = (24 - 4y) / 3
The beauty of the equalization method lies in this moment. Since both of these expressions are equal to x, that means they must also be equal to each other! It's like saying if A = C and B = C, then A = B. Make sense? So, we can confidently set the two expressions equal to each other, creating a new equation that looks like this:
10 - 2y = (24 - 4y) / 3
Ta-da! We've eliminated x and now we have a single equation with only one variable, y. This is a huge step forward because we know how to solve equations with just one variable. This new equation is our ticket to finding the value of y, which will then lead us to the value of x. But before we start solving, let's just take a moment to appreciate what we've done here. We've taken a system of two equations with two unknowns and transformed it into a single equation with one unknown. This is the power of the equalization method! It allows us to simplify complex problems and break them down into manageable steps. Now, it's time to roll up our sleeves and solve for y. We'll need to use our algebraic skills to manipulate this equation and isolate y. But don't worry, we'll take it one step at a time, and you'll see that it's totally doable. Remember, the key is to stay organized and focused. Keep your eye on the goal, which is to get y all by itself on one side of the equation. And with a little bit of effort, we'll be there in no time!
Step 3: Solve for y
Okay, let's dive into solving for y in our new equation:
10 - 2y = (24 - 4y) / 3
The first thing we want to do is get rid of that fraction. Fractions can be a bit messy to work with, so let's multiply both sides of the equation by 3. This will cancel out the denominator on the right side:
3 * (10 - 2y) = 3 * [(24 - 4y) / 3]
This simplifies to:
30 - 6y = 24 - 4y
Now we have a nice, clean equation without any fractions. The next step is to get all the y terms on one side of the equation and all the constant terms on the other side. Let's start by adding 6y to both sides:
30 - 6y + 6y = 24 - 4y + 6y
This gives us:
30 = 24 + 2y
Now, let's subtract 24 from both sides to isolate the y term:
30 - 24 = 24 + 2y - 24
This simplifies to:
6 = 2y
Finally, to solve for y, we divide both sides by 2:
6 / 2 = 2y / 2
And we get:
y = 3
Woohoo! We've found the value of y! It's equal to 3. This is a major accomplishment, guys. We're one step closer to solving the entire system of equations. But we're not done yet. We still need to find the value of x. But now that we know y, that's going to be a piece of cake. Remember, the key to solving for y was to carefully manipulate the equation, performing the same operations on both sides to maintain balance. We got rid of the fraction, combined like terms, and isolated y. And now, we have our answer. So, let's take a moment to celebrate our success and then move on to the final step: finding x.
Step 4: Substitute y into One of the Original Equations to Solve for x
Alright, we've nailed down the value of y – it's 3! Now comes the fun part: using that knowledge to find x. This step is all about substitution, which is a fancy way of saying we're going to plug the value of y (which is 3) into one of our original equations. It's like we're taking a piece of the puzzle we've already solved and using it to unlock the next piece. So, which equation should we choose? Well, the beauty of it is, it doesn't matter! We can pick either of the original equations, and we'll still get the correct value for x. But to make things as easy as possible, let's choose the simpler one:
x + 2y = 10
This equation looks a bit less complicated than the other one, so it'll probably be easier to work with. Now, we substitute y = 3 into this equation. This means we replace the y in the equation with the number 3:
x + 2 * 3 = 10
See how we just swapped out the y for its value? Now we have an equation with only one unknown, x, which we know how to solve. Let's simplify this equation:
x + 6 = 10
Now, to isolate x, we simply subtract 6 from both sides:
x + 6 - 6 = 10 - 6
This gives us:
x = 4
Boom! We've found it! The value of x is 4. We've successfully solved for both x and y. This is a huge accomplishment, guys. We took a system of equations, used the equalization method, and found the values that satisfy both equations. We're like algebraic detectives, solving the mystery of the unknowns. But before we declare our case closed, there's one more important thing we should do: check our solution.
Step 5: Check Your Solution
We've found our potential solution: x = 4 and y = 3. But before we throw a party, it's super important to check if these values actually work in both of our original equations. This is like the final exam for our solution – we want to make sure it passes with flying colors! So, let's take our first equation:
x + 2y = 10
And substitute x = 4 and y = 3:
4 + 2 * 3 = 10
Let's simplify:
4 + 6 = 10
10 = 10
Woohoo! It checks out! Our solution works in the first equation. But we're not done yet. We need to make sure it also works in the second equation:
3x + 4y = 24
Let's substitute again:
3 * 4 + 4 * 3 = 24
Simplify:
12 + 12 = 24
24 = 24
Double woohoo! It checks out in the second equation too! This means our solution is absolutely correct. x = 4 and y = 3 is the solution to the system of equations. Checking our solution is such a crucial step because it ensures that we haven't made any mistakes along the way. It's like having a safety net – it catches us if we've slipped up somewhere. And it gives us the confidence to know that our answer is accurate. So, always, always, always check your solution! It's the mark of a true math whiz.
Conclusion
Guys, we did it! We successfully solved the system of equations x + 2y = 10 and 3x + 4y = 24 using the equalization method. We found that x = 4 and y = 3. We walked through each step, from isolating a variable to setting the expressions equal, solving for y, substituting to find x, and finally, checking our solution. I hope this step-by-step guide has made the equalization method clear and understandable. Remember, practice makes perfect! The more you use this method, the more comfortable you'll become with it. So, go out there and tackle some more systems of equations! You've got this! And remember, math can be fun, especially when you're solving problems like a pro. Keep practicing, keep exploring, and keep those math muscles strong!