Notebook & Pen Price: Solving A System Of Equations

Hey guys! Ever wondered how to solve a problem where you have multiple unknowns? Today, we're diving into a classic math problem that involves finding the price of notebooks and pens using a system of equations. It might sound intimidating, but trust me, it's like cracking a code! We'll break it down step by step, so you'll be a system-solving pro in no time. So, let's jump into this mathematical adventure and figure out the cost of these everyday items. Understanding these concepts is super useful, not just for school, but also for real-life situations where you need to compare prices and make smart decisions. Remember, math is not just about numbers; it's about problem-solving, and that's a skill that will serve you well in all aspects of life.

Setting Up the Equations: The Key to Unlocking the Solution

So, our main question here is: What are the prices of a notebook and a pen if 2 notebooks and 3 pens cost R$ 26.00, and 4 notebooks and 2 pens cost R$ 40.00? To solve this, let's first translate the word problem into mathematical equations. This is a crucial step in solving any word problem. It's like translating a sentence from one language to another – you need to understand the grammar and vocabulary of math! We'll use variables to represent the unknowns. Let's say the price of one notebook is 'x' and the price of one pen is 'y'. Now, we can rewrite the given information as two equations:

• Equation 1: 2x + 3y = 26 (This represents the cost of 2 notebooks and 3 pens)
• Equation 2: 4x + 2y = 40 (This represents the cost of 4 notebooks and 2 pens)

These two equations form a system of linear equations. A system of equations is simply a set of two or more equations with the same variables. Our goal is to find the values of 'x' and 'y' that satisfy both equations simultaneously. Think of it like finding the perfect combination that fits both scenarios. There are several methods to solve systems of equations, but we'll focus on the elimination method in this case. Setting up the equations correctly is half the battle won! It's like laying the foundation for a building – if the foundation is solid, the rest of the structure will be strong too. So, let's make sure we understand how we got these equations before moving on to the next step. Feel free to pause and re-read this section if needed. Got it? Awesome! Let's move on to the next stage: eliminating a variable.

The Elimination Method: Our Strategy for Success

Now that we have our system of equations, we can use the elimination method to solve for our variables. The elimination method is a technique where we manipulate the equations in such a way that one of the variables cancels out, leaving us with a single equation with only one variable. This makes it much easier to solve! The basic idea is to multiply one or both equations by a constant so that the coefficients of one of the variables are opposites. In our case, we can multiply Equation 1 by -2:

• -2 * (2x + 3y) = -2 * 26
• -4x - 6y = -52

Now we have a modified Equation 1: -4x - 6y = -52. Notice that the coefficient of 'x' in this modified equation is the opposite of the coefficient of 'x' in Equation 2 (4x). This is exactly what we wanted! Now, we can add this modified equation to Equation 2:

  -4x - 6y = -52
+ 4x + 2y = 40
----------------
  0x - 4y = -12

The 'x' terms cancel out, and we're left with a simple equation: -4y = -12. This is a huge step forward! We've eliminated one variable and now we can easily solve for 'y'. The elimination method is like a clever trick that simplifies the problem. It might seem a bit confusing at first, but with practice, you'll become a master of elimination! The key is to look for opportunities to create opposite coefficients. Sometimes you might need to multiply both equations by different constants to achieve this. But don't worry, it's all about practice and understanding the underlying concept. So, let's move on and solve for 'y'. We're getting closer to the solution!

Solving for 'y': Finding the Price of a Pen

We've reached the point where we have a simple equation: -4y = -12. To solve for 'y' (which represents the price of a pen), we need to isolate 'y' on one side of the equation. We can do this by dividing both sides of the equation by -4:

-4y / -4 = -12 / -4
y = 3

So, we've found that y = 3. This means the price of one pen is R$ 3.00! Yay! We've cracked the code for one of the unknowns. Solving for a variable is like peeling away the layers of an onion – each step gets you closer to the core. In this case, the core is the value of 'y'. Remember, when you divide both sides of an equation by the same number, you're maintaining the balance. It's like keeping both sides of a scale equal. Now that we know the price of a pen, we're halfway there! But we still need to find the price of a notebook ('x'). Don't worry, we're on a roll! The next step is to substitute this value of 'y' back into one of our original equations to solve for 'x'. This is like using a piece of the puzzle to find the missing pieces. So, let's move on and find out the price of a notebook.

Solving for 'x': Unveiling the Cost of a Notebook

Now that we know y = 3 (the price of a pen), we can substitute this value into either Equation 1 or Equation 2 to solve for 'x' (the price of a notebook). Let's use Equation 1: 2x + 3y = 26. Substituting y = 3, we get:

2x + 3(3) = 26
2x + 9 = 26

Now we need to isolate 'x'. First, subtract 9 from both sides of the equation:

2x + 9 - 9 = 26 - 9
2x = 17

Next, divide both sides by 2:

2x / 2 = 17 / 2
x = 8.5

So, we've found that x = 8.5. This means the price of one notebook is R$ 8.50! Awesome! We've successfully solved for both 'x' and 'y'. Substituting the value of one variable into another equation is a powerful technique. It's like using a bridge to cross from one side of a river to the other. By substituting, we connected the value of 'y' to the equation and were able to solve for 'x'. Remember, when solving equations, it's important to perform the same operation on both sides to maintain the balance. This ensures that the equation remains true. Now that we have both 'x' and 'y', we've cracked the entire code! Let's summarize our findings and choose the correct answer.

The Final Answer: Putting It All Together

We've solved the system of equations and found that:

• x = 8.5 (Price of a notebook is R$ 8.50)
• y = 3 (Price of a pen is R$ 3.00)

Now, let's look at the options provided in the question:

• (A) R$ 8.00 e R$ 2.00
• (B) R$ 10.00 e R$ 2.00
• (C) R$ 9.00 e R$ 3.00
• (D) R$ 6.00 e R$ 4.00

None of the options exactly match our solution (R$ 8.50 and R$ 3.00). However, option (C) R$ 9.00 e R$ 3.00 is the closest to our answer. It's possible that there was a slight rounding error in the problem or the options provided. In a real-world scenario, you might want to double-check the problem statement or clarify with the person who gave you the problem. But based on our calculations, option (C) is the most reasonable answer. Putting it all together is like assembling the pieces of a puzzle. Each step we took – setting up the equations, eliminating a variable, solving for 'y', and solving for 'x' – was a piece of the puzzle. And now, we have the complete picture! It's important to remember that math is not just about getting the right answer; it's about the process of problem-solving. We learned valuable skills in this process, such as translating word problems into equations, using the elimination method, and substituting values. These skills will be useful in many other areas of math and in life in general. So, congratulations! You've successfully solved a system of equations and found the price of notebooks and pens. Keep practicing, and you'll become a math whiz in no time!

Real-World Applications: Why This Matters

You might be thinking, "Okay, I solved a math problem, but how is this useful in real life?" Well, solving systems of equations has many practical applications! Think about it: any time you need to compare costs, plan a budget, or analyze data, you might encounter situations where you need to solve for multiple unknowns. For example:

• Budgeting: Imagine you're planning a trip and need to figure out how much you can spend on accommodation and activities. You have a total budget and know the average cost of each. You can set up a system of equations to determine the optimal amount to spend on each category.
• Business: Businesses use systems of equations to analyze sales data, determine pricing strategies, and manage inventory. For example, a store might want to figure out how many units of two different products they need to sell to reach a certain revenue target.
• Science and Engineering: Scientists and engineers use systems of equations to model complex systems, such as electrical circuits, chemical reactions, and fluid dynamics.

So, the skills you've learned in solving this problem are not just abstract mathematical concepts; they are valuable tools that can help you make informed decisions in many areas of your life. Understanding systems of equations is like having a superpower – you can see through the complexity and find the underlying relationships. And the more you practice, the stronger your superpower will become! So, keep exploring the world of math, and you'll be amazed at how much it can help you in your daily life. This is the beauty of mathematics; it's not just about numbers, it's about understanding the world around us!

Practice Makes Perfect: Keep Honing Your Skills

Now that you've successfully navigated this notebook and pen problem, the key to mastering systems of equations is practice! The more you work through different examples, the more comfortable you'll become with the process. Think of it like learning a new language – the more you practice speaking and writing, the more fluent you'll become. There are plenty of resources available to help you practice:

• Textbooks: Your math textbook likely has many examples and practice problems related to systems of equations.
• Online Resources: Websites like Khan Academy and Mathway offer lessons, practice problems, and step-by-step solutions.
• Worksheets: You can find printable worksheets online or create your own problems.

Try working through different types of problems, such as those involving different scenarios or using different methods of solving (like substitution). Challenge yourself to solve problems without looking at the solutions, and then check your work to see where you can improve. And don't be afraid to ask for help! If you're struggling with a particular concept, reach out to your teacher, a tutor, or a classmate. Learning math is a collaborative process, and we can all learn from each other. Remember, every mistake is an opportunity to learn and grow. So, embrace the challenges, keep practicing, and you'll become a system-solving superstar! The world of mathematics is vast and fascinating, and the more you explore it, the more you'll discover its beauty and power. So, keep your curiosity alive, and never stop learning!