Solving For Uniform Costs Using A System Of Equations
Introduction
In the realm of mathematics, we often encounter real-world scenarios that can be elegantly solved using systems of equations. These systems, composed of two or more equations with shared variables, provide a powerful framework for deciphering unknown quantities. This article delves into a compelling problem involving a soccer team's uniform expenses, where we'll leverage the power of systems of equations to determine the cost of each jersey and pair of shorts. Understanding these concepts is crucial for anyone involved in budgeting, resource allocation, or simply making informed financial decisions. This problem not only highlights the practical applications of algebra but also underscores the importance of analytical thinking in everyday life. Let's embark on this mathematical journey and unravel the intricacies of this problem, demonstrating how mathematical tools can illuminate real-world scenarios and provide precise solutions.
Problem Statement: Decoding the Uniform Costs
A soccer team's financial officer faces a common budgeting challenge: determining the individual costs of team uniforms. The team initially placed an order for 12 jerseys and 12 pairs of shorts, incurring a total expense of $156. Subsequently, as the team expanded, they needed to order an additional 4 jerseys and 6 pairs of shorts, resulting in a further outlay of $62. The central question we aim to address is: What is the cost of each jersey, and what is the cost of each pair of shorts? This problem perfectly exemplifies how systems of equations can be employed to solve real-world financial puzzles. By translating the given information into mathematical expressions, we can systematically unravel the unknowns and arrive at a precise solution. This approach not only provides a numerical answer but also enhances our understanding of how algebraic tools can be applied to practical scenarios, making budgeting and financial planning more transparent and efficient.
Setting Up the System of Equations: Translating Reality into Math
To solve this problem, we must first translate the given information into a system of linear equations. This involves representing the unknown quantities—the cost of a jersey and the cost of a pair of shorts—with variables. Let's denote the cost of each jersey as x and the cost of each pair of shorts as y. The initial order of 12 jerseys and 12 pairs of shorts, totaling $156, can be expressed as the equation: 12x + 12y = 156. Similarly, the subsequent order of 4 jerseys and 6 pairs of shorts, costing $62, can be represented by the equation: 4x + 6y = 62. Together, these two equations form a system that accurately models the given scenario. This step is crucial because it transforms a word problem into a mathematical framework that can be systematically analyzed and solved. The beauty of this approach lies in its ability to distill complex information into manageable algebraic expressions, paving the way for a clear and concise solution.
Solving the System of Equations: Unveiling the Costs
Now that we have established our system of equations:
- 12x + 12y = 156
- 4x + 6y = 62
We can employ various methods to solve for x and y. One common approach is the elimination method, which involves manipulating the equations to eliminate one variable, allowing us to solve for the other. Let's multiply the second equation by -3 to make the coefficients of x in both equations opposites:
-3 * (4x + 6y) = -3 * 62
This simplifies to:
-12x - 18y = -186
Now we have the following equations:
- 12x + 12y = 156
- -12x - 18y = -186
Adding these two equations together eliminates x:
(12x + 12y) + (-12x - 18y) = 156 + (-186)
This simplifies to:
-6y = -30
Dividing both sides by -6, we find:
y = 5
Now that we know the cost of a pair of shorts (y) is $5, we can substitute this value back into either of the original equations to solve for x. Let's use the first equation:
12x + 12(5) = 156
Simplifying:
12x + 60 = 156
Subtracting 60 from both sides:
12x = 96
Dividing both sides by 12, we find:
x = 8
Therefore, the cost of each jersey (x) is $8, and the cost of each pair of shorts (y) is $5. This systematic approach demonstrates the power of algebraic techniques in solving practical problems, providing a clear and concise pathway to the solution.
Conclusion: Practical Applications of Systems of Equations
In conclusion, by setting up and solving a system of equations, we have successfully determined that each jersey costs $8, and each pair of shorts costs $5. This exercise demonstrates the practical application of systems of equations in real-world scenarios, particularly in budgeting and financial planning. The ability to translate real-world problems into mathematical models allows for precise and efficient solutions, aiding in informed decision-making. Whether it's a sports team managing their uniform expenses or a business allocating resources, the principles of algebra provide a powerful toolkit for navigating complex financial landscapes. This exploration not only reinforces the importance of mathematical literacy but also highlights the elegance and utility of algebraic methods in everyday life. Understanding and applying these concepts can significantly enhance our ability to analyze situations, make informed choices, and achieve desired outcomes in various aspects of life and work.
