Introduction
In this intriguing mathematical puzzle, we're presented with a scenario involving the purchase of fidgets and Pop Its. Fidgets, those small, often colorful gadgets designed to provide tactile stimulation and relieve stress, come at a cost of $3 each. Pop Its, the endlessly poppable silicone toys that have taken the world by storm, are priced at $4 apiece. The challenge lies in figuring out how many of each item were bought, given that a total of 20 Fidgets and Pop Its were purchased for a combined cost of $75. This problem is a perfect example of how a system of equations can be used to solve real-world scenarios. Let's embark on this mathematical journey, where we'll explore how to translate the given information into a system of equations, ultimately paving the way to determine the number of fidgets and Pop Its acquired.
Defining the Variables: The Foundation of Our Equations
Before we can construct our equations, it's crucial to define our variables clearly. In this case, we're told that $x$ represents the number of fidgets purchased. This is a great starting point. Now, let's introduce another variable, $y$, to represent the number of Pop Its bought. By establishing these variables, we've laid the foundation for translating the word problem into mathematical expressions. The beauty of algebra lies in its ability to represent unknown quantities with symbols, allowing us to manipulate them and uncover their values. With our variables defined, we're ready to move on to the next step: constructing the equations that capture the relationships described in the problem.
Crafting the Equations: Translating Words into Math
Now comes the heart of the problem-solving process: transforming the given information into a system of equations. Remember, we have two key pieces of information: the total number of items purchased and the total cost. Let's tackle the first piece of information: a total of 20 Fidgets and Pop Its were bought. Since $x$ represents the number of fidgets and $y$ represents the number of Pop Its, we can express this relationship as an equation: x + y = 20. This equation elegantly captures the fact that the sum of fidgets and Pop Its equals 20. Now, let's turn our attention to the second piece of information: the total cost of $75. We know that fidgets cost $3 each, so the total cost of fidgets is $3x$. Similarly, Pop Its cost $4 each, making the total cost of Pop Its $4y$. The combined cost of fidgets and Pop Its is $75, which we can express as the equation 3x + 4y = 75. This equation represents the financial aspect of the problem, linking the quantities of fidgets and Pop Its to their respective costs and the overall expenditure.
The System of Equations: A Mathematical Representation
With both equations in hand, we can now present the system of equations that encapsulates the entire problem:
- x + y = 20 (Total number of items)
- 3x + 4y = 75 (Total cost)
This system of equations is a powerful tool. It's a compact and precise representation of the relationships between the number of fidgets, the number of Pop Its, and the total cost. The beauty of a system of equations lies in its ability to capture multiple constraints simultaneously. In this case, we have two equations and two unknowns, which means we have a good chance of finding a unique solution. The next step would be to solve this system using techniques like substitution, elimination, or graphing. However, the question asks us to identify the system of equations, which we have successfully done. This system of equations serves as a roadmap, guiding us towards the solution to our puzzle.
Solving the System (Optional): Unveiling the Number of Fidgets and Pop Its
While the problem only asks for the system of equations, let's take it a step further and explore how we might solve it. Solving the system will reveal the actual number of fidgets and Pop Its purchased. There are several methods we could use, but let's focus on the substitution method. The first equation, x + y = 20, is relatively simple to manipulate. We can solve it for one variable in terms of the other. Let's solve for $x$: x = 20 - y. Now we have an expression for $x$ that we can substitute into the second equation.
Substitute x = 20 - y into the second equation, 3x + 4y = 75, to get:
3(20 - y) + 4y = 75
Now, we have a single equation with one variable, $y$. Let's simplify and solve for $y$:
60 - 3y + 4y = 75
y = 15
We've found that y = 15, which means 15 Pop Its were purchased. Now, we can substitute this value back into either of our original equations to find $x$. Let's use the equation x + y = 20:
x + 15 = 20
x = 5
So, we've determined that 5 fidgets were purchased. This confirms that our system of equations was not only correctly formulated but also led us to the correct solution. We have successfully unraveled the mystery of the fidgets and Pop Its!
Importance of Systems of Equations: A Powerful Problem-Solving Tool
This exercise highlights the power and versatility of systems of equations. They are not just abstract mathematical concepts; they are practical tools that can be used to solve real-world problems. From calculating mixtures in chemistry to determining optimal pricing strategies in business, systems of equations provide a framework for analyzing situations with multiple interconnected variables. The ability to translate word problems into mathematical equations is a crucial skill in various fields. By mastering this skill, we can tackle complex problems with clarity and precision. Systems of equations empower us to model real-world scenarios, make informed decisions, and gain a deeper understanding of the relationships that govern our world. This example, involving fidgets and Pop Its, serves as a tangible illustration of the practical applications of this fundamental mathematical concept.
Conclusion: The Elegance of Mathematical Modeling
In conclusion, we've successfully navigated the world of fidgets and Pop Its using the power of systems of equations. We've seen how to translate a word problem into a mathematical representation, capturing the essential relationships between variables. The system of equations we derived, x + y = 20 and 3x + 4y = 75, encapsulates the core information of the problem: the total number of items and the total cost. By solving this system (which we did as an optional step), we uncovered the specific quantities of fidgets and Pop Its purchased. This journey underscores the elegance and practicality of mathematical modeling. Systems of equations are not just abstract constructs; they are powerful tools that empower us to solve real-world problems, make informed decisions, and gain insights into the interconnectedness of the world around us. The fidgets and Pop Its example serves as a reminder that mathematics is not confined to textbooks; it is a vibrant and essential part of our daily lives.