Notebooks And Paper Costs: A Math Problem Solved

Hey guys! Ever found yourself scratching your head over a math problem that seems like it's written in another language? Well, I get it! Math word problems can be tricky, but trust me, once you break them down, they're like puzzles waiting to be solved. Today, we're going to tackle a classic: figuring out the cost of notebooks and paper when you're given a couple of different purchase scenarios. Think of it as becoming a math detective – cool, right?

Unraveling the Problem: Israel's Stationery Shopping Spree

So, here's the situation: Israel, our math adventurer, went on two separate shopping trips to the same stationery store. On the first trip, he shelled out $360 for six packs of paper and five notebooks. Fast forward to his second visit, and he paid $120 for just one pack of paper and two notebooks. Our mission, should we choose to accept it, is to figure out the individual cost of a notebook and a pack of paper. Sounds like a mission impossible? Nah, we've got this! This is a classic system of equations problem disguised as a real-world scenario. These types of problems are super common, not just in math class, but also in everyday life. Imagine you're trying to figure out the best deal on bulk snacks or comparing phone plans. The same problem-solving skills apply! So, let's dive in and see how we can crack this case. We'll use a step-by-step approach to make sure we understand every twist and turn. Are you ready to become a math whiz? Let's get started!

Step 1: Translating Words into Math - The Power of Variables

The first thing we need to do is turn this wordy problem into something math can understand. Think of it as translating from English to Math-lish! We do this by using variables. Variables are like placeholders – they stand in for the things we don't know yet. In our case, we don't know the cost of a single pack of paper or a single notebook. So, let's give them names. We'll let 'x' represent the cost of one pack of paper. Seems fair, right? And we'll let 'y' represent the cost of one notebook. Now we have our secret code! This is a crucial step because it transforms our problem from a confusing jumble of words into clear, actionable math. It's like giving each unknown a name tag so we can keep track of them. Once we have these variables defined, we can start building our equations. Equations are the sentences of math, and they'll help us express the relationships between the things Israel bought. So, with our variables in hand, we're ready to move on to the next step: writing those equations! Remember, the key is to break down the information we have and express it in a way that math can work with. We're on our way to solving this puzzle, one variable at a time. So keep that detective hat on, and let's keep going!

Step 2: Building the Equations - Math Sentences

Now comes the fun part – turning the information about Israel's purchases into mathematical equations! Remember those variables we defined? This is where they shine. Let's look at the first shopping trip. Israel bought six packs of paper, and each pack costs 'x' dollars. So, the total cost of the paper is 6 * x, or simply 6x. He also bought five notebooks, and each notebook costs 'y' dollars. That's a total of 5 * y, or 5y. And we know that the grand total for this trip was $360. So, we can write our first equation: 6x + 5y = 360. Ta-da! We've translated a whole sentence from the problem into math. Feels pretty powerful, doesn't it? Now, let's tackle the second shopping trip. This time, Israel bought one pack of paper (that's 1x, or just x) and two notebooks (that's 2y). The total cost for this trip was $120. So, our second equation is: x + 2y = 120. Boom! We've done it again. We now have two equations, and they perfectly capture the information from our word problem. These equations are like the key to unlocking the mystery of the paper and notebook costs. We call this a system of equations because we have multiple equations working together to solve for multiple unknowns. It's like a team effort in the math world. With these two equations, we're ready to move on to the next step: choosing a method to solve them. There are several ways to crack this code, and we'll explore one of the most common and effective methods next. So, let's keep building on our progress, one equation at a time!

Step 3: Solving the System - Choosing Your Weapon

Alright, we've got our two equations: 6x + 5y = 360 and x + 2y = 120. Now the real fun begins – actually solving for 'x' and 'y'! There are a couple of popular ways to tackle a system of equations, but one of the most versatile is called the substitution method. Think of it like this: we're going to substitute one variable in terms of the other, kind of like swapping out pieces in a puzzle until everything fits. The first thing we need to do is pick one of our equations and solve for one of the variables. It doesn't matter which one you choose, but it's often easiest to pick the equation where a variable has a coefficient of 1 (that's the number in front of the variable). In our case, the second equation (x + 2y = 120) has a lonely 'x', which makes it a perfect candidate. So, let's solve that equation for 'x'. To do that, we simply subtract 2y from both sides of the equation. This gives us: x = 120 - 2y. Awesome! We've now expressed 'x' in terms of 'y'. This is the key to the substitution method. We're going to take this expression for 'x' and substitute it into our other equation. It's like we're taking what we learned from the second equation and using it to help us solve the first. So, in our first equation (6x + 5y = 360), we're going to replace 'x' with '120 - 2y'. This might sound a little complicated, but trust me, it's just a clever way to reduce our problem to a single equation with a single variable. And once we have that, we're golden! So, let's take a deep breath and get ready for the substitution. We're about to make some serious progress in solving this mystery!

Step 4: Substitution in Action - Replacing the Unknown

Okay, let's put our substitution plan into action! Remember, we have x = 120 - 2y, and we're going to plug that into our first equation, which is 6x + 5y = 360. This means wherever we see an 'x' in the first equation, we're going to replace it with the expression '120 - 2y'. So, the equation becomes: 6(120 - 2y) + 5y = 360. See what we did there? We've swapped out 'x' for its equivalent expression. Now, we have a single equation with only one variable, 'y'. This is a huge step forward! It means we're one step closer to finding the cost of a notebook. But before we can solve for 'y', we need to simplify this equation. This involves a little bit of algebra magic – specifically, the distributive property. The distributive property tells us that we need to multiply the 6 outside the parentheses by each term inside the parentheses. So, 6 * 120 is 720, and 6 * -2y is -12y. Our equation now looks like this: 720 - 12y + 5y = 360. We're almost there! Now we just need to combine like terms and isolate 'y'. We have a -12y and a +5y, which combine to give us -7y. So, our equation simplifies to: 720 - 7y = 360. We're in the home stretch! 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. We'll do this in the next step, and then we'll finally be able to reveal the value of 'y'. So, keep your eyes on the prize, and let's keep going!

Step 5: Solving for 'y' - Unveiling the Notebook Cost

We've reached a critical point in our math adventure! Our equation is 720 - 7y = 360. To isolate 'y', we need to get rid of the 720 on the left side. We can do this by subtracting 720 from both sides of the equation. Remember, whatever we do to one side, we have to do to the other to keep the equation balanced. So, 720 - 7y - 720 = 360 - 720. This simplifies to -7y = -360. We're getting closer! Now, we just need to get 'y' by itself. It's currently being multiplied by -7, so to undo that, we'll divide both sides of the equation by -7. This gives us: y = -360 / -7. A negative divided by a negative is a positive, so we have: y = 51.43 (approximately). This is super exciting! We've found the value of 'y', which represents the cost of one notebook. So, one notebook costs approximately $51.43. High five! We've solved for one of our unknowns. But our mission isn't complete yet. We still need to find the cost of a pack of paper, which is represented by 'x'. But don't worry, we're not starting from scratch. We can use the value of 'y' that we just found to help us solve for 'x'. We'll do that in the next step, and then we'll have the complete solution to our stationery mystery. So, let's keep up the momentum and move on to the final piece of the puzzle!

Step 6: Finding 'x' - The Paper Pack Price

We've conquered the notebook cost, and now it's time to uncover the price of a pack of paper! Remember, we found that y = 51.43 (approximately). We can use this value and plug it back into one of our original equations to solve for 'x'. The easiest equation to use is x + 2y = 120 because 'x' is already pretty much isolated. So, we substitute 51.43 for 'y' in this equation: x + 2(51.43) = 120. Now, we simplify. 2 * 51.43 is approximately 102.86. So, our equation becomes: x + 102.86 = 120. To get 'x' by itself, we subtract 102.86 from both sides of the equation: x = 120 - 102.86. This gives us: x = 17.14 (approximately). We've done it! We've found the value of 'x', which represents the cost of one pack of paper. So, a pack of paper costs approximately $17.14. Give yourselves a round of applause! We've successfully navigated the world of systems of equations and solved for both 'x' and 'y'. But before we declare victory, there's one final step we should always take: checking our answer. This is like the detective double-checking their work to make sure they've got the right suspect. We'll do that in the next step, just to be 100% sure we've cracked the case.

Step 7: Checking Our Work - Math Detective Double-Check

Alright, math detectives, it's time to put on our critical thinking caps and double-check our solution. We found that a pack of paper (x) costs approximately $17.14, and a notebook (y) costs approximately $51.43. To check our work, we're going to plug these values back into our original equations and see if they hold true. Let's start with the first equation: 6x + 5y = 360. Substituting our values, we get: 6(17.14) + 5(51.43) = 360. Now, we do the math. 6 * 17.14 is approximately 102.84, and 5 * 51.43 is approximately 257.15. So, our equation becomes: 102.84 + 257.15 = 360. Adding those two numbers together, we get approximately 359.99, which is very close to 360! That's a good sign. Now, let's check our second equation: x + 2y = 120. Substituting our values, we get: 17.14 + 2(51.43) = 120. Again, we do the math. 2 * 51.43 is approximately 102.86. So, our equation becomes: 17.14 + 102.86 = 120. Adding those numbers together, we get exactly 120! Woohoo! Both of our equations hold true with our values for 'x' and 'y'. This means we've successfully solved the system of equations and found the correct costs for the paper and notebooks. We've officially cracked the case! Give yourselves a pat on the back – you've earned it. You've not only solved a math problem, but you've also learned a valuable problem-solving skill that can be applied to all sorts of situations in life. So, the next time you're faced with a tricky problem, remember the steps we took today: define your variables, build your equations, choose a solution method, and always check your work. You've got this!

Conclusion: Mission Accomplished!

So, there you have it, guys! We've successfully navigated the world of systems of equations and figured out that a pack of paper costs approximately $17.14, and a notebook costs approximately $51.43. We broke down a seemingly complex problem into manageable steps, used variables to represent unknowns, built equations to express relationships, and employed the substitution method to find our answers. And most importantly, we checked our work to ensure accuracy. This is the power of math – it gives us the tools to solve real-world problems and make informed decisions. Whether you're comparing prices at the grocery store, planning a budget, or even designing a building, the problem-solving skills you learn in math class are invaluable. So, keep practicing, keep exploring, and never be afraid to tackle a challenge. You might just surprise yourself with what you can accomplish. And remember, math isn't just about numbers and equations; it's about thinking critically, creatively, and logically. It's about becoming a problem-solving superhero in your own life. So, go forth and conquer, math adventurers! The world is full of puzzles waiting to be solved, and you've got the skills to crack them all. Until next time, keep those brains buzzing and those pencils moving!