Shirts & Pants Cost: Solve The Math Puzzle!
Have you ever found yourself staring at a bill, trying to figure out the individual price of each item? It's like a mini-math mystery, and today, we're diving into one of those scenarios. Let's break down this problem step-by-step, so you can confidently solve similar puzzles in the future. This is a classic example of a system of equations, and it's super useful in everyday life. So, grab your thinking caps, guys, and let's get started!
The Shirt and Pants Problem: A Mathematical Mystery
Okay, so here's the deal: María went shopping and snagged two cool shirts and a stylish pair of pants, spending a total of 22 euros. Meanwhile, Pedro, not to be outdone, picked up three shirts and two pairs of pants, shelling out 39 euros. The big question is: how much does each shirt and each pair of pants cost individually? This is where our math skills come into play, and we'll unravel this mystery together.
Setting Up the Equations: Our Detective Tools
To solve this, we'll use a system of equations. Think of equations as our detective tools, helping us uncover the hidden prices. First, we need to assign variables. Let's say:
- x = the price of one shirt
- y = the price of one pair of pants
Now, we can translate the information from the problem into equations:
- María's purchase: 2x + y = 22
- Pedro's purchase: 3x + 2y = 39
See? We've turned a word problem into a set of mathematical statements. These equations are our clues, and now we need to use some algebraic techniques to solve for x and y.
Solving the System: Cracking the Code
There are a couple of ways we can solve this system of equations: substitution or elimination. Let's go with the elimination method this time. The goal is to eliminate one of the variables (either x or y) by manipulating the equations. We can do this by multiplying one or both equations by a constant so that the coefficients of one variable are opposites. Then, when we add the equations together, that variable will cancel out.
Looking at our equations:
- 2x + y = 22
- 3x + 2y = 39
We can eliminate y by multiplying the first equation by -2. This will give us a -2y term, which will cancel out with the +2y term in the second equation. So, let's do that:
- -2 * (2x + y) = -2 * 22
- -4x - 2y = -44
Now we have a new set of equations:
- -4x - 2y = -44
- 3x + 2y = 39
Time to add these equations together! Notice how the -2y and +2y terms will cancel each other out:
- (-4x + 3x) + (-2y + 2y) = -44 + 39
- -x = -5
To solve for x, we simply multiply both sides by -1:
- x = 5
Eureka! We've found the price of one shirt: 5 euros. Now, we need to find the price of the pants.
Finding the Pants Price: Completing the Puzzle
Now that we know x = 5, we can substitute this value into either of our original equations to solve for y. Let's use the first equation, 2x + y = 22:
- 2 * (5) + y = 22
- 10 + y = 22
Subtract 10 from both sides:
- y = 12
And there you have it! The price of one pair of pants is 12 euros.
The Solution: Mystery Solved!
So, after all our mathematical detective work, we've discovered that each shirt costs 5 euros, and each pair of pants costs 12 euros. We successfully used a system of equations to solve this real-world problem. Wasn't that fun, guys? Remember, this same approach can be used to solve all sorts of problems involving multiple unknowns. The key is to break it down step-by-step, set up your equations carefully, and choose the method that works best for you. Keep practicing, and you'll become a math whiz in no time!
Why This Matters: Real-World Applications of Systems of Equations
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