Modeling Lunch Account Balances: Equations For Spending

Hey guys! Today, we're diving into a fun little problem about modeling real-life situations with equations. Think of it as turning a story into math! We've got Enrique and Maya, both managing their lunch money. Let's break down their situations and see how we can represent them using equations. This will not only help us understand their spending habits but also give us a powerful tool for solving similar problems in the future. So, grab your thinking caps, and let's get started!

Understanding the Problem

To really nail this, let's first make sure we really understand what's going on. Enrique starts with $50 in his lunch account and spends $5 each day. That's our first character and our first set of numbers. Maya, on the other hand, starts with $46 and spends $4 per day. We've got two people, two starting amounts, and two different spending rates. The key here is to figure out how to write equations that show how much money each of them has left in their accounts after a certain number of days. We're looking for equations that link the number of days (which we can call 'x') to the amount of money they have left (which we can call 'y'). Think of 'x' as the input (number of days) and 'y' as the output (remaining balance). Got it? Awesome, let's move on to the next part where we start building those equations.

Defining the Variables

Before we jump into the equations, let's clearly define our variables. This is super important because it helps us keep track of what each part of the equation means. We're going to use 'x' to represent the number of days that have passed. So, if we're talking about 5 days, then x = 5. Simple, right? Now, 'y' is going to represent the amount of money left in the account after 'x' days. This is what we're trying to find – the balance remaining after a certain amount of spending. Using variables like this is a fundamental part of algebra, and it allows us to express relationships in a concise and clear way. It's like a secret code that helps us translate real-world scenarios into mathematical language. By defining 'x' and 'y', we've laid the groundwork for building our equations, making the whole process much smoother. So, with our variables in place, let's see how we can use them to represent Enrique and Maya's spending habits!

Building the Equations

Now for the fun part – turning these scenarios into equations! Let's start with Enrique. He begins with $50, which is our starting point. He's spending $5 every day, so that amount is going to decrease his balance. We can represent this decrease as subtracting $5 for each day. If 'x' is the number of days, then he spends a total of 5 * x dollars. So, the amount of money Enrique has left, 'y', can be written as: y = 50 - 5x. See how we took the starting amount and subtracted the total spending? That's the core idea. Now, let's tackle Maya. She starts with $46, and she spends $4 each day. Using the same logic, her spending can be represented as 4 * x dollars. So, the equation for Maya's remaining balance, 'y', is: y = 46 - 4x. We've now successfully created two equations that model their spending habits. These equations are like mini-programs that tell us exactly how much money each person has left after any given number of days. Cool, huh? Let's dive deeper and see what these equations actually tell us about their spending patterns.

Enrique's Equation: y = 50 - 5x

Let's break down Enrique's equation: y = 50 - 5x. The 50 is super important because it's his initial balance – the amount he starts with. Think of it as the foundation of his lunch money castle. The 5x part represents the total amount he spends over 'x' days. The minus sign tells us this amount is being subtracted from his starting balance. So, for every day ('x') that passes, he's losing $5. Now, 'y' is the result – it's the amount he has left after spending. This equation is a linear equation, which means if we were to graph it, it would be a straight line. The slope of that line (which is -5) tells us how quickly his balance is decreasing. The y-intercept (which is 50) tells us where the line starts on the y-axis. Understanding these parts helps us visualize Enrique's spending. For example, if x = 1 (one day), y = 50 - 5(1) = $45. He has $45 left. If x = 10 (ten days), y = 50 - 5(10) = $0. He's broke! This equation is a powerful tool for predicting his balance at any point in time. So, let's see if we can do the same thing with Maya's equation!

Maya's Equation: y = 46 - 4x

Alright, let's dissect Maya's spending habits, represented by the equation: y = 46 - 4x. Just like with Enrique, the number 46 is the key – it's Maya's starting balance. This is the treasure chest she begins with. Now, the 4x part represents how much she spends in total. The fact that it's subtracted tells us her balance is going down each day. The variable 'x' is still our number of days, and 'y' is the grand total she has remaining. So, for each day that passes, she's shelling out $4. This equation, like Enrique's, is also a linear equation. If we graphed it, we'd see another straight line, but this one would have a slightly different slope (-4) and a different y-intercept (46). The slope tells us she's spending money at a slower rate than Enrique, and the y-intercept is her initial stash. Let's throw in some numbers to see it in action. If x = 1 (one day), y = 46 - 4(1) = $42. She has $42 left. If x = 10 (ten days), y = 46 - 4(10) = $6. She still has some money! This equation lets us peer into her financial future, predicting her balance after any number of days. By comparing it to Enrique's equation, we can even see who's going to run out of money first. Equations are like magic mirrors, revealing hidden patterns and predictions!

The Answer

So, after all that equation-building and analyzing, what's the final answer? The equations that model this situation are:

• Enrique: y = 50 - 5x
• Maya: y = 46 - 4x

These equations tell the whole story of their lunch account spending. They show us how their balances change over time and give us a way to predict their future spending habits. Remember, 'x' is the number of days, and 'y' is the amount of money left. You can plug in any number of days for 'x' and calculate how much money they'll have left. This is the power of using equations to model real-world scenarios. We've taken a word problem and turned it into a clear, concise mathematical representation. You guys nailed it! We've tackled the problem, built the equations, and understood what they mean. Now you've got the tools to solve similar problems and impress your friends with your math skills. Keep practicing, and you'll become equation-building pros in no time!