Bicycle Price: Solving Hugo's Payment Plan Equation
Hey guys! Let's dive into a fun math problem today. We're going to figure out how much a bicycle costs based on a payment plan. Hugo is buying a bike from his brother and paying him off over time. We have an equation that models how much Hugo still owes, and we're going to use that equation to find the bike's original price. This is a cool example of how math can be used in everyday situations, and it's a great way to flex those problem-solving muscles! So, grab your thinking caps, and let's get started!
Understanding the Problem
The problem states that Hugo agreed to pay his brother $2 per week for a bicycle. The equation provided, y - 10 = -2(x - 10), models the amount of money Hugo owes for the bike. Here, x represents the number of weeks, and y represents the amount of money Hugo still needs to pay. The question we need to answer is: how much did the bicycle originally cost?
To solve this, we need to understand what the equation is telling us. The equation is in point-slope form, which is a useful way to represent a linear relationship. Let's break down what each part of the equation means in the context of Hugo's bicycle purchase:
- y: This variable represents the amount of money Hugo still owes his brother after a certain number of weeks. So, if y is a large number, Hugo still has a significant debt to pay. If y is a small number, Hugo is close to paying off the bicycle. And if y is zero, Hugo has paid off the entire amount!
- x: This variable represents the number of weeks that have passed since Hugo started paying for the bicycle. The more weeks that pass, the more Hugo pays off, and the lower the amount he owes should become.
- -2: This is the slope of the line. In this context, the slope represents the rate at which Hugo is paying off the bicycle. The negative sign indicates that the amount he owes is decreasing over time. The value '2' specifically tells us that Hugo is paying $2 per week, which aligns with the information given in the problem.
- (x - 10): This part of the equation relates the number of weeks (x) to a specific point on the line. The '10' inside the parentheses is part of the point-slope form and helps us identify a specific week number to consider.
- (y - 10): Similar to the above, this part relates the amount owed (y) to a specific point on the line. The '10' here is linked to the other '10' in the equation and helps us define a known point on the line representing Hugo's payment plan.
By carefully dissecting the equation and understanding the meaning of each component, we can start to see how to use this information to determine the bicycle's original cost. We need to find a point where we can definitively say how much the bike cost, and the equation gives us the tools to do just that!
Solving for the Original Price
Okay, let's get down to solving this problem step-by-step. Remember, our goal is to find the original price of the bicycle. To do this, we'll use the equation y - 10 = -2(x - 10) and think about what it represents at different points in time.
First, let's simplify the equation. We can distribute the -2 on the right side:
y - 10 = -2x + 20
Next, let's isolate y by adding 10 to both sides:
y = -2x + 30
Now, we have the equation in slope-intercept form (y = mx + b), which is often easier to work with. In this form, we can see that:
- -2 is the slope (as we discussed earlier).
- 30 is the y-intercept. The y-intercept is the value of y when x is 0. In our context, x is the number of weeks, so x = 0 means the beginning, before Hugo has made any payments. The value of y at this point represents the initial amount Hugo owed, which is the original price of the bicycle!
So, based on this, it seems like the bicycle originally cost $30. However, let's verify this by thinking about another point in time. We know Hugo pays $2 per week. Let's figure out how many weeks it takes him to pay off the bike.
To find this, we need to find when y (the amount owed) is equal to 0. So, we set the equation to 0:
0 = -2x + 30
Now, let's solve for x:
2x = 30
x = 15
This means it takes Hugo 15 weeks to pay off the bicycle. Now, let's think about this logically. If Hugo pays $2 per week for 15 weeks, he pays a total of:
$2/week * 15 weeks = $30
This confirms our earlier finding! The bicycle originally cost $30.
We've solved the problem in two ways: by interpreting the y-intercept of the equation and by calculating the total amount paid over time. Both methods give us the same answer, which gives us confidence in our solution. Great job, guys! We've successfully used algebra to solve a real-world problem.
Why This Equation Works
Now that we've figured out the price of the bicycle, let's take a step back and really analyze why the equation y - 10 = -2(x - 10) works in modeling Hugo's debt. Understanding the underlying principles will not only solidify your understanding of this problem but also equip you with the tools to tackle similar situations in the future. It's not just about getting the answer; it's about understanding the process!
The key here is the point-slope form of a linear equation, which is generally written as:
y - y1 = m(x - x1)
Where:
- m is the slope of the line (the rate of change).
- (x1, y1) is a specific point on the line.
Our equation, y - 10 = -2(x - 10), is in this exact form. We've already identified that the slope, m, is -2, representing the $2 decrease in debt each week. But what about the point (x1, y1)? In our case, it's (10, 10). This means that at week 10 (x = 10), Hugo owes $10 (y = 10).
Think about this for a moment. At week 10, Hugo still owes $10. This is a specific, known point in Hugo's payment plan. The equation is built around this point and uses the slope to calculate the amount owed at any other week. This is the magic of the point-slope form – it lets us define a line based on a single point and its direction (slope).
Let's consider how the equation changes as x increases (as weeks pass). For every increase of 1 in x, the term (x - 10) increases by 1. Since this term is multiplied by -2, the entire right side of the equation decreases by 2. This decrease is then reflected in y (the amount owed), which also decreases by 2. This perfectly mirrors Hugo's payment plan: for every week that passes, he owes $2 less.
But why the '10's? Why this specific point? Well, the problem could have used any point on the line to define the equation. The point (10, 10) was likely chosen because it represents a convenient intermediate state in Hugo's payment plan. It's not the starting point (when he owes the full amount), and it's not the ending point (when he owes nothing). It's a point somewhere in the middle that helps to anchor the equation.
If we had a different point on the line, we could write a different, but equivalent, equation. For example, we already calculated that at week 15, Hugo owes $0. So, the point (15, 0) is also on the line. We could write the equation in point-slope form using this point:
y - 0 = -2(x - 15)
This equation looks different, but if you simplify it, you'll find that it's equivalent to our original equation. This highlights a key concept: there are often multiple ways to represent the same mathematical relationship.
By understanding the point-slope form and how it relates to Hugo's payment plan, we gain a much deeper understanding of the problem. We can see how the equation captures the essence of the situation and allows us to accurately calculate the amount owed at any point in time. This kind of conceptual understanding is crucial for truly mastering mathematics and applying it to real-world problems.
Real-World Applications of Linear Equations
Okay, guys, we've nailed down the bicycle problem, but let's zoom out for a minute and think about the bigger picture. Why is understanding these kinds of equations so important? The truth is, linear equations like the one we used to model Hugo's debt are everywhere in the real world. They pop up in finance, physics, engineering, and countless other fields. Learning to recognize and work with them is a seriously valuable skill.
Think about it: any situation where there's a constant rate of change can be modeled with a linear equation. Here are just a few examples:
- Loan Payments: Just like Hugo's bicycle, most loans (car loans, mortgages, student loans) follow a payment plan that can be modeled with a linear equation. The slope represents your monthly payment, and the y-intercept represents the original loan amount. Understanding these equations can help you predict how long it will take to pay off a loan and how much interest you'll end up paying.
- Savings and Investments: If you're saving money at a consistent rate, or if you're earning a fixed interest rate on an investment, you can use a linear equation to track your progress. The slope represents your savings rate or the interest earned, and the y-intercept represents your initial savings or investment.
- Distance, Rate, and Time: The classic formula distance = rate * time is a linear equation! If you're driving at a constant speed, the distance you travel is linearly related to the time you spend driving. This concept is used in navigation, logistics, and many other applications.
- Cost and Revenue: Businesses often use linear equations to model their costs and revenues. For example, the cost of producing a certain number of items might include a fixed cost (like rent) and a variable cost per item (like materials). The revenue from selling those items might be linearly related to the number of items sold. By analyzing these equations, businesses can make decisions about pricing, production levels, and profitability.
- Physics: Linear equations are fundamental in physics. For example, the motion of an object moving at a constant velocity can be described by a linear equation. Similarly, Ohm's Law, which relates voltage, current, and resistance in an electrical circuit, is a linear equation.
The key takeaway here is that the skills we used to solve Hugo's bicycle problem – understanding slope, intercepts, and the point-slope form – are broadly applicable. By mastering these concepts, you're not just learning how to solve textbook problems; you're building a foundation for understanding and solving real-world challenges. So, keep practicing, keep exploring, and keep applying your mathematical knowledge to the world around you! You'll be surprised at how often it comes in handy.
Conclusion: Math is Everywhere!
Alright, guys, we've reached the end of our mathematical journey for today, and what a journey it's been! We started with a simple question about the price of a bicycle and ended up exploring the power of linear equations and their applications in the real world. We not only figured out that Hugo's bicycle cost $30, but we also delved deep into why the equation worked, how it relates to the point-slope form, and how linear equations show up in all sorts of unexpected places.
I hope you've realized that math isn't just about memorizing formulas and solving abstract problems. It's a powerful tool for understanding and interacting with the world around us. Whether you're figuring out a payment plan, planning a road trip, or making business decisions, the principles we've discussed today can be incredibly valuable.
Remember, the key to mastering math isn't just about getting the right answer; it's about understanding why the answer is right. It's about connecting the dots between abstract concepts and concrete situations. It's about developing a problem-solving mindset that you can apply to any challenge you encounter.
So, keep asking questions, keep exploring, and keep challenging yourselves. The world is full of mathematical puzzles just waiting to be solved. And who knows? Maybe the next time you're faced with a real-world problem, you'll think back to Hugo's bicycle and realize that you already have the tools you need to find the solution. Keep up the awesome work, guys! You've got this!