Solve For Pens And Styluses: A Math Puzzle!

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Hey guys! Ever been stumped by a math problem that feels like a real-life shopping scenario? Well, today, we're diving into a classic algebra problem that's all about figuring out the cost of pens and styluses. It's super useful because it shows how math can help us solve everyday situations. Let's break it down and make it easy to understand! The problem goes like this: We know that 7 styluses and 6 pens cost 235 lei. Also, we know that 2 styluses and 3 pens cost a certain amount of money. Our mission? To find out the individual price of one pen and one stylus. Ready to put on your detective hats and solve this math mystery?

Setting Up the Equations

Alright, the first step in solving any good algebra problem is to translate the words into math. This means we need to create equations. Don't worry; it's easier than it sounds! Let's use 'x' to represent the cost of a stylus and 'y' to represent the cost of a pen. From the problem, we have two key pieces of information that we can turn into equations.

  • Equation 1: 7 styluses and 6 pens cost 235 lei. This translates to: 7x + 6y = 235.
  • Equation 2: 2 styluses and 3 pens cost an unknown amount (let's call it 'z' for now). This translates to: 2x + 3y = z.

So, we have two equations, but we need to find the value of 'x' and 'y'. To get the value of 'z' is optional, but in order to solve the first two variables, we will need to use the first equation. We'll use the first equation to solve for the variables, let's dive in.

Solving the Equations: The Elimination Method

There are a few ways to solve a system of equations, but we'll use the elimination method here. The goal is to eliminate one of the variables (either 'x' or 'y') so that we can solve for the other. Here’s how we do it:

  1. Make the Coefficients Match: Look at the coefficients of 'y' in both equations (6 and 3). The easiest way to eliminate 'y' is to make these coefficients equal in magnitude but opposite in sign. Multiply the second equation by -2. This gives us a -6y, which will cancel out the 6y in the first equation. So, the second equation becomes -4x - 6y = -2z.
  2. Add the Equations: Now, add the modified second equation to the first equation: (7x + 6y) + (-4x - 6y) = 235 - 2z. This simplifies to 3x = 235 - 2z.
  3. Solve for x: To find the value of 'x', we'll first need to calculate the value of 'z'. To do that we will isolate the value of 'y'. Using Equation 2, we'll multiply by 2, so we can cancel the 'y's. This gives us 4x + 6y = 2z. Subtract this new equation with the first one, like so: 7x + 6y - (4x + 6y) = 235 - 2z, which simplifies to 3x = 235 - 2z.
  4. Solve for z: Next, we'll subtract Equation 2 with Equation 1, like so: (7x + 6y) - (2x + 3y) = 235 - z, which simplifies to 5x + 3y = 235 - z. Because the value of 'z' is unknown, we will replace it with the value of the first equation. Then, we'll subtract the second equation with the first one again: (7x + 6y) - (2x + 3y) = 235 - (2x + 3y), which simplifies to 5x + 3y = 235 - 2x - 3y. If we move all the variables to the left side, we'll have 7x + 6y = 235. We now have a system of equations, with which we can find the value of x and y. Finally, we'll subtract the second equation by 2 to get 4x + 6y = 2z. If we subtract the first equation, we'll have 3x = 235 - 2z. We can now isolate the variables.

Finding the Cost of a Pen and a Stylus

Let's say that the total cost of the second equation is 100 lei, which gives us the following system of equations:

  • 7x + 6y = 235
  • 2x + 3y = 100

Then, follow the steps:

  1. Make the Coefficients Match: Look at the coefficients of 'y' in both equations (6 and 3). The easiest way to eliminate 'y' is to make these coefficients equal in magnitude but opposite in sign. Multiply the second equation by -2. This gives us a -6y, which will cancel out the 6y in the first equation. So, the second equation becomes -4x - 6y = -200.
  2. Add the Equations: Now, add the modified second equation to the first equation: (7x + 6y) + (-4x - 6y) = 235 - 200. This simplifies to 3x = 35.
  3. Solve for x: Divide both sides by 3 to find the value of 'x': x = 35 / 3, so x ≈ 11.67. This means each stylus costs approximately 11.67 lei.
  4. Solve for y: Now that we know the value of 'x', we can substitute it back into either of the original equations to solve for 'y'. Let's use the second original equation 2x + 3y = 100. Substitute x = 11.67, which gives us 2 * 11.67 + 3y = 100, so 23.34 + 3y = 100. Subtract 23.34 from both sides: 3y = 76.66. Divide by 3: y = 76.66 / 3, so y ≈ 25.55. This means each pen costs approximately 25.55 lei.

Therefore, using our assumption that 2 styluses and 3 pens cost 100 lei, we have approximately the values of 11.67 lei for the stylus and 25.55 lei for the pen. If we know the total cost, we can easily calculate the exact price.

Conclusion: Math in Action!

And there you have it, guys! We successfully solved the problem and figured out the approximate cost of a stylus and a pen. This demonstrates how algebra can be used to solve everyday problems, making it easier to understand the world around us. So next time you’re shopping and see prices, remember that math is there to help you navigate it all. Keep practicing, and you’ll become a math whiz in no time!