Solve Equations: Find Tube & Cardboard Costs!

Hey guys! Ever found yourself staring at a math problem and feeling totally lost? Don't worry, we've all been there. Today, we're going to tackle a fun word problem that involves setting up equations to figure out the cost of some supplies. Imagine you're building a model, and you need tubes and cardboard. One group bought 3 tubes and 2 cardboard pieces for $1000, while another group got 2 tubes and 1 cardboard piece for $650. The big question is: how much does each tube and each cardboard piece cost? Sounds intriguing, right? Let’s break it down step by step and make it super clear. Math can be like a puzzle, and we're about to put all the pieces together!

Setting up the Equations

Okay, the first step in solving any word problem is to translate the words into mathematical language. It's like learning a new code, but trust me, it’s easier than it sounds! Our main keywords here are tubes and cardboard, and we need to figure out their individual costs. So, let's assign variables to these unknowns. We'll let 'x' represent the cost of one tube, and 'y' will represent the cost of one cardboard piece. Think of 'x' and 'y' as our mystery boxes that we're going to open.

Now, let’s look at the information we have. The first group bought 3 tubes and 2 cardboard pieces for $1000. We can write this as an equation: 3x + 2y = 1000. See how we turned the words into a math sentence? The '3x' means three times the cost of a tube, and '2y' means two times the cost of a cardboard piece. The plus sign simply adds them together, and the total cost is $1000.

The second group bought 2 tubes and 1 cardboard piece for $650. We can write this as another equation: 2x + y = 650. Again, '2x' is two times the cost of a tube, and 'y' is the cost of one cardboard piece. This time, they spent $650 in total. We now have two equations that describe our situation. This is what we call a system of equations, and it’s our key to unlocking the answer.

Why is this important? Well, each equation gives us a relationship between the cost of tubes and cardboard. By having two equations, we can use different methods to solve for our two unknowns, 'x' and 'y'. We're essentially creating a mathematical map that will lead us to the right answer. Remember, the goal is to find the exact value of 'x' and 'y', so we know how much each item costs. Keep this in mind as we move to the next step!

Solving the System of Equations: The Elimination Method

Alright, we've got our two equations: 3x + 2y = 1000 and 2x + y = 650. Now comes the fun part: solving them! There are a few ways to tackle this, but today we'll use the elimination method. This method is super handy because it helps us get rid of one variable, making it easier to solve for the other. Think of it as strategically canceling things out to simplify the problem.

The main idea behind the elimination method is to multiply one or both equations by a constant so that the coefficients (the numbers in front of the variables) of either 'x' or 'y' are opposites. This way, when we add the equations together, one of the variables will disappear. Let's focus on eliminating 'y' in our case. Notice that in the first equation, the coefficient of 'y' is 2, and in the second equation, it's 1. To make them opposites, we can multiply the entire second equation by -2. This will give us -2y, which is the opposite of 2y.

So, let's do that: -2 * (2x + y) = -2 * 650. This simplifies to -4x - 2y = -1300. Now we have a modified second equation. Our system of equations now looks like this:

• 3x + 2y = 1000
• -4x - 2y = -1300

See how the coefficients of 'y' are now opposites? Perfect! Now we can add the two equations together. When we add the left sides, we get (3x + 2y) + (-4x - 2y). The '2y' and '-2y' cancel each other out, leaving us with 3x - 4x, which simplifies to -x. On the right side, we have 1000 + (-1300), which equals -300. So, our combined equation is -x = -300. To solve for 'x', we simply multiply both sides by -1, which gives us x = 300.

We’ve found our first answer! The cost of one tube (x) is $300. That wasn't so bad, was it? Now that we know 'x', we can plug it back into one of our original equations to solve for 'y'.

Finding the Value of y: Substituting x

Okay, now that we've cracked the code for 'x' (the cost of a tube), it's time to find 'y' (the cost of a cardboard piece). We know that x = 300, and we have our two original equations: 3x + 2y = 1000 and 2x + y = 650. The easiest way to find 'y' is to use the substitution method. This means we'll replace 'x' in one of the equations with its value, which is 300.

Let’s choose the second equation, 2x + y = 650, because it looks a bit simpler. We substitute x = 300 into this equation: 2 * 300 + y = 650. Now we have an equation with only one variable, 'y', which makes it much easier to solve. Multiply 2 by 300, and we get 600 + y = 650.

To isolate 'y', we need to get rid of the 600. We can do this by subtracting 600 from both sides of the equation: 600 + y - 600 = 650 - 600. This simplifies to y = 50. We've found 'y'! The cost of one cardboard piece is $50. See how substituting the value of 'x' helped us find 'y'? It's like connecting the dots to reveal the final picture.

Now we know the cost of both a tube and a cardboard piece. A tube costs $300, and a cardboard piece costs $50. But before we celebrate our victory, let’s make sure our answers are correct. We need to check our solution in both original equations.

Verifying the Solution

Alright, we’ve found that x = 300 (the cost of a tube) and y = 50 (the cost of a cardboard piece). But math is like a detective game – we need to verify our clues to make sure we’ve got the right answer. This is where checking our solution comes in. It’s a crucial step because it ensures we haven't made any mistakes along the way.

We’ll plug our values of 'x' and 'y' into both of our original equations and see if they hold true. If they do, we know we’ve solved the problem correctly. If not, it means we need to go back and check our work. Let's start with the first equation: 3x + 2y = 1000. We substitute x = 300 and y = 50: 3 * 300 + 2 * 50 = 1000. Now let's simplify: 900 + 100 = 1000. And guess what? 1000 = 1000. Our solution works for the first equation! That’s a good sign.

Now let’s check the second equation: 2x + y = 650. We substitute x = 300 and y = 50: 2 * 300 + 50 = 650. Let’s simplify: 600 + 50 = 650. And again, 650 = 650. Our solution works for the second equation as well! This means we’ve nailed it. We’ve successfully solved the system of equations.

Why is this step so important? Imagine if we made a small mistake in our calculations. Without checking, we’d end up with the wrong answers, and our model-building budget would be all messed up! Verifying our solution gives us the confidence that we’ve done the math correctly and that our answers make sense in the context of the problem. It's like the final seal of approval on our mathematical masterpiece.

Conclusion: You've Cracked the Code!

Woo-hoo! You’ve made it to the end, guys! We’ve taken a real-world problem, turned it into a system of equations, solved it using the elimination and substitution methods, and even verified our solution. That's a lot of math power right there! We started with the question of how much tubes and cardboard cost, and now we know the answer: a tube costs $300, and a cardboard piece costs $50.

This main problem solving process is a fantastic skill to have, not just for math class, but for all sorts of situations in life. Whether you’re planning a budget, figuring out ingredient amounts for a recipe, or even solving a puzzle, the ability to break down a problem into smaller parts and solve it step by step is incredibly valuable.

Remember, the key to success in math (and in life!) is to take things one step at a time. Don't be afraid to make mistakes – they're part of the learning process. And always, always check your work to make sure you’re on the right track. You’ve got this! So, the next time you encounter a tricky problem, remember the steps we’ve covered today, and go crack that code!

Keep practicing, keep learning, and keep shining! You've got the tools now to tackle all sorts of equation-solving adventures. And who knows? Maybe you’ll even build the most awesome model ever with your newfound math skills! Until next time, happy solving!