Solving A System Of Equations: Sum And Difference Problem
Hey guys! Ever stumbled upon a math problem that seems like a word puzzle? This one's a classic: "The sum of two numbers is 82. Their difference is 24. Write a system of equations that describes this situation. Solve by elimination to find the two numbers." Sounds tricky? Don't sweat it! We're going to break it down, step by step, in a way that's super easy to follow. So, let's jump right in and become equation-solving pros!
Setting Up the Equations: The Key to Unlocking the Problem
First off, let's talk about how we translate words into math. This is a crucial skill, not just for math class, but for all sorts of problem-solving situations in life. When you see phrases like "the sum of two numbers," think addition. "Their difference"? That's subtraction. We're on a mission to turn these phrases into neat and tidy equations.
So, how do we do it? We use variables! Let's call our two mystery numbers 'x' and 'y'. This is super common in algebra – we use letters to stand in for the unknowns we're trying to find. Now, let's look at the information we're given. The problem says, "The sum of two numbers is 82." That means if we add our two numbers, x and y, we should get 82. We can write that as an equation: x + y = 82. See? We're already making progress!
Next up, we have "Their difference is 24." Remember, difference means subtraction. So, if we subtract our two numbers, x and y, we should get 24. That gives us our second equation: x - y = 24. Bam! We've got ourselves a system of equations:
x + y = 82
x - y = 24
This is the heart of the problem. We've taken the words and turned them into math. This system of equations is like a map that's going to lead us to our treasure – the two numbers we're trying to find. This is where the magic starts to happen. Remember, each equation represents a relationship between x and y. To solve for x and y individually, we need to use both equations together. Think of it like two pieces of a puzzle that fit perfectly to reveal the answer.
Why is setting up equations so important? Well, without the right equations, we're basically wandering in the dark. The equations are our guide, our roadmap, our decoder ring for this math mystery. If we mess up the equations, the rest of the solution will be off too. So, take your time with this step, read the problem carefully, and make sure your equations accurately reflect the information you're given. It's the foundation for everything that follows.
Think of it like building a house. You wouldn't start putting up walls before you had a solid foundation, right? Setting up the equations correctly is like laying that foundation for our math problem. Once we have that solid base, we can confidently move on to the next step: solving the system.
Solving by Elimination: A Powerful Technique
Alright, we've got our system of equations. Now for the fun part: solving it! We're going to use a method called "elimination," which is a super handy trick for solving systems of equations. The basic idea behind elimination is to get rid of one of the variables (either x or y) by adding the two equations together. Sounds a bit like magic, right? Let's see how it works.
Take a good look at our system:
x + y = 82
x - y = 24
Notice anything special? Check out the 'y' terms. We have a '+y' in the first equation and a '-y' in the second equation. This is perfect! When we add these two equations together, the 'y' terms are going to cancel each other out. It's like they're enemies, and they're going to annihilate each other in a glorious mathematical showdown! This is the beauty of elimination.
So, let's do it. Let's add the two equations together, term by term:
- x + x = 2x
- y + (-y) = 0 (The 'y's are gone!)
- 82 + 24 = 106
Putting it all together, we get a brand new equation: 2x = 106. See how much simpler that is? We've eliminated the 'y' variable, and now we just have one equation with one unknown.
Now, how do we solve for 'x'? Easy peasy! We just need to isolate 'x'. Remember, 2x means 2 times x. To undo the multiplication, we do the opposite operation: division. We'll divide both sides of the equation by 2:
2x / 2 = 106 / 2
x = 53
Boom! We've found 'x'. One of our mystery numbers is 53. But we're not done yet! We still need to find 'y'. Finding 'x' is like finding one piece of the puzzle, but we need the other piece to complete the picture.
So, what do we do now? Well, we can take the value of 'x' that we just found (x = 53) and plug it back into either one of our original equations. It doesn't matter which one you choose; you'll get the same answer for 'y' either way. Let's use the first equation, x + y = 82, because it looks a little simpler. We'll substitute 53 for x:
53 + y = 82
Now we have another simple equation to solve. To isolate 'y', we need to get rid of the 53. Since it's being added to 'y', we'll do the opposite: subtract 53 from both sides:
53 + y - 53 = 82 - 53
y = 29
Awesome! We've found 'y'. Our second mystery number is 29. We've cracked the code!
Verifying the Solution: Double-Checking Our Work
We've found our two numbers, x = 53 and y = 29. But how do we know if we're right? Well, this is where verification comes in. It's like the final boss level of our math game. We want to make sure our solution works perfectly.
The key to verifying our solution is to plug our values for 'x' and 'y' back into our original equations. If both equations hold true, then we know we've nailed it. If not, then we need to go back and check our work. Think of it like proofreading a paper – you want to catch any mistakes before you turn it in.
Let's start with the first equation: x + y = 82. We'll substitute 53 for x and 29 for y:
53 + 29 = 82
Is this true? Yes! 53 + 29 does indeed equal 82. So far, so good. We've passed the first test!
Now let's try the second equation: x - y = 24. Again, we'll substitute 53 for x and 29 for y:
53 - 29 = 24
Is this true? Absolutely! 53 - 29 equals 24. We've passed the second test!
Since our values for 'x' and 'y' satisfy both equations, we can confidently say that our solution is correct. We've conquered the problem! This is the satisfaction of a job well done. Verifying our solution isn't just a formality; it's a crucial step that gives us peace of mind. It's like having a safety net – it catches us if we've made a mistake along the way.
Think of it like this: you wouldn't launch a rocket into space without checking all the systems, right? Verifying our solution is like checking the systems of our math problem to make sure everything is working perfectly. It's a sign of a careful and thorough mathematician!
So, the next time you solve a system of equations, don't skip the verification step. It's the final piece of the puzzle, and it ensures that your hard work pays off with a correct answer. It's the difference between being pretty sure you're right and knowing you're right.
Conclusion: Math Mystery Solved!
And there you have it! We've successfully solved a system of equations using the elimination method. We started with a word problem, translated it into mathematical equations, solved for our unknowns, and verified our solution. You're basically math superheroes now!
We found that the two numbers are 53 and 29. This wasn't just about finding the right answer; it was about understanding the process, the logic, the art of problem-solving. These are skills that will serve you well in all areas of life.
Remember, math isn't just about numbers and formulas; it's about thinking critically, breaking down problems, and finding creative solutions. So, keep practicing, keep exploring, and keep challenging yourself. You've got this! And if you ever get stuck, remember the steps we went through today: set up the equations, solve using elimination, and verify your solution. You'll be solving math mysteries like a pro in no time!
