Solving Systems Of Equations To Find Sandwich Prices

Jul 16, 2025 by ADMIN 53 views

Solving Sandwich Prices A System of Equations Approach

Introduction

In this article, we'll explore a classic problem-solving scenario using a system of equations. The problem involves Dan and Mario purchasing sandwiches at a shop, with different quantities of tuna and vegetarian sandwiches leading to different total costs. Our goal is to determine the cost of each type of sandwich by setting up and analyzing a system of equations. This problem is a great example of how algebra can be applied to everyday situations, and it highlights the power of mathematical modeling in understanding and solving real-world problems. Understanding how to formulate and solve systems of equations is a fundamental skill in mathematics, with applications extending far beyond simple word problems. From economics and finance to engineering and computer science, the ability to represent relationships between variables and solve for unknowns is crucial. This article will not only walk you through the solution to the sandwich problem but also provide insights into the broader context and significance of systems of equations. By the end of this exploration, you'll have a clearer understanding of how algebraic techniques can be used to unravel complex situations and arrive at precise solutions. The beauty of mathematics lies in its ability to transform seemingly intricate problems into manageable steps, and this sandwich scenario perfectly illustrates that principle. So, let's dive into the world of equations and discover how they can help us decipher the cost of tuna and vegetarian sandwiches.

Problem Statement

Dan and Mario's Sandwich Purchase: Dan and Mario visited a sandwich shop. Dan bought 2 tuna sandwiches and 5 vegetarian sandwiches, spending a total of $39. Mario, on the other hand, bought 4 tuna sandwiches and 3 vegetarian sandwiches for $37. The challenge is to determine which system of equations can be used to solve for the cost of each kind of sandwich. This is a classic problem that lends itself perfectly to the use of algebraic techniques, specifically systems of equations. By translating the given information into mathematical expressions, we can create a model that represents the relationships between the quantities and costs involved. This approach not only allows us to find the solution but also provides a structured way to think about similar problems in the future. The key to solving this problem lies in identifying the unknowns, defining variables to represent them, and then formulating equations based on the given information. Each equation will represent a relationship between the number of sandwiches purchased and the total cost, allowing us to create a system of equations that can be solved simultaneously. This method is not only applicable to this specific scenario but can be generalized to a wide range of problems involving multiple variables and constraints. So, let's embark on the journey of transforming this word problem into a mathematical model and unraveling the cost of those delicious sandwiches.

Defining Variables

To begin solving this problem, the first crucial step is to define variables that will represent the unknowns we are trying to find. In this case, we want to determine the cost of a tuna sandwich and the cost of a vegetarian sandwich. Let's use the following variables:

Let x represent the cost of one tuna sandwich.
Let y represent the cost of one vegetarian sandwich.

By assigning these variables, we are essentially creating a symbolic representation of the unknown quantities, which allows us to translate the word problem into mathematical expressions. This process of abstraction is fundamental to algebra and enables us to manipulate and solve equations more effectively. The choice of variables is arbitrary, but it's often helpful to use letters that are suggestive of the quantities they represent, such as x and y in this case. Once we have defined our variables, we can proceed to translate the information given in the problem statement into equations that involve these variables. Each equation will represent a relationship between the number of sandwiches purchased and the total cost, providing us with a system of equations that can be solved to find the values of x and y. This step of defining variables is not just a matter of notation; it's a critical step in framing the problem in a way that allows us to apply algebraic techniques and arrive at a solution. So, with our variables defined, we are now ready to move on to the next step: formulating the equations.

Formulating the Equations

Now that we have defined our variables, we can translate the information provided in the problem statement into mathematical equations. Recall that Dan bought 2 tuna sandwiches and 5 vegetarian sandwiches for $39, and Mario bought 4 tuna sandwiches and 3 vegetarian sandwiches for $37. We can represent these purchases as follows:

Dan's purchase: 2x + 5y = 39
Mario's purchase: 4x + 3y = 37

In the first equation, 2x represents the cost of the 2 tuna sandwiches Dan bought, and 5y represents the cost of the 5 vegetarian sandwiches. The sum of these costs is equal to the total amount Dan spent, which is $39. Similarly, in the second equation, 4x represents the cost of Mario's 4 tuna sandwiches, and 3y represents the cost of his 3 vegetarian sandwiches, totaling $37. These two equations form a system of linear equations, which is a set of two or more equations that involve the same variables. Solving this system will give us the values of x and y, which represent the cost of a tuna sandwich and a vegetarian sandwich, respectively. The process of translating word problems into mathematical equations is a crucial skill in algebra and allows us to apply algebraic techniques to solve real-world problems. The ability to identify the relationships between quantities and express them in a concise mathematical form is essential for problem-solving in various fields, from science and engineering to economics and finance. So, with our system of equations formulated, we are now ready to explore different methods for solving it and finding the cost of each type of sandwich.

System of Equations

The system of equations that represents the given information is:

2x + 5y = 39
4x + 3y = 37

This system consists of two linear equations, each involving the variables x and y. The first equation, 2x + 5y = 39, represents Dan's purchase, where 2x is the cost of the tuna sandwiches and 5y is the cost of the vegetarian sandwiches. The second equation, 4x + 3y = 37, represents Mario's purchase, with 4x being the cost of the tuna sandwiches and 3y being the cost of the vegetarian sandwiches. To solve this system of equations, we need to find values for x and y that satisfy both equations simultaneously. In other words, we are looking for the point of intersection of the two lines represented by these equations on a graph. There are several methods we can use to solve this system, including substitution, elimination, and graphing. Each method has its own advantages and disadvantages, and the choice of method often depends on the specific form of the equations and the personal preference of the solver. Regardless of the method used, the goal remains the same: to find the values of x and y that make both equations true. This system of equations is a powerful representation of the problem, allowing us to use algebraic techniques to find the solution. It encapsulates the relationships between the quantities and costs involved, providing a framework for solving the problem in a systematic and rigorous way. So, with our system of equations in hand, we are now ready to delve into the various methods for solving it and uncovering the cost of those sandwiches.

Solving the System of Equations

There are a couple of common methods for solving systems of equations like the one we've established: substitution and elimination. Let's explore the elimination method for this particular problem, as it can be quite efficient.

Elimination Method

The elimination method involves manipulating the equations in such a way that one of the variables is eliminated when the equations are added or subtracted. To do this, we need to multiply one or both equations by a constant so that the coefficients of one of the variables are opposites. In our system:

2x + 5y = 39
4x + 3y = 37

Notice that the coefficient of x in the second equation (4) is twice the coefficient of x in the first equation (2). This suggests that we can eliminate x by multiplying the first equation by -2 and then adding the equations together.

Multiply the first equation by -2:
```
-2(2x + 5y) = -2(39)
-4x - 10y = -78
```
Now, add the modified first equation to the second equation:
```
(-4x - 10y) + (4x + 3y) = -78 + 37
-7y = -41
```
Solve for y:
```
y = -41 / -7
y ≈ 5.86
```
So, the cost of a vegetarian sandwich is approximately $5.86.
Now that we have the value of y, we can substitute it into either of the original equations to solve for x. Let's use the first equation:
```
2x + 5(5.86) = 39
2x + 29.3 = 39
2x = 9.7
x = 4.85
```
Therefore, the cost of a tuna sandwich is approximately $4.85.

The elimination method provides a systematic way to solve systems of equations by strategically manipulating the equations to eliminate variables. This approach is particularly useful when the coefficients of one of the variables are multiples of each other, as it simplifies the process of finding the solution. By following these steps, we have successfully determined the cost of each type of sandwich, demonstrating the power of algebraic techniques in solving real-world problems. This method not only provides a solution but also enhances our understanding of the relationships between the variables and the underlying structure of the problem. So, with the values of x and y determined, we have effectively unraveled the mystery of the sandwich prices.

Solution

Based on the elimination method, we have found the following solutions:

The cost of a tuna sandwich (x) is approximately $4.85.
The cost of a vegetarian sandwich (y) is approximately $5.86.

This solution satisfies both equations in the system, meaning that these prices for the sandwiches are consistent with the information given about Dan and Mario's purchases. To verify this, we can substitute these values back into the original equations and check if they hold true. For Dan's purchase, 2x + 5y = 2(4.85) + 5(5.86) = 9.7 + 29.3 = 39, which matches the total cost of $39. For Mario's purchase, 4x + 3y = 4(4.85) + 3(5.86) = 19.4 + 17.58 = 36.98, which is approximately $37 (the slight difference is due to rounding). This verification step is crucial in ensuring the accuracy of the solution and confirming that we have correctly applied the algebraic techniques. The solution not only provides the prices of the sandwiches but also demonstrates the power of systems of equations in solving real-world problems. By translating the word problem into a mathematical model, we were able to use algebraic methods to arrive at a precise and meaningful solution. This approach can be applied to a wide range of problems involving multiple variables and constraints, making it a valuable tool in various fields, from science and engineering to economics and finance. So, with the solution in hand, we have not only solved the sandwich problem but also gained a deeper appreciation for the versatility and effectiveness of mathematical problem-solving.

Conclusion

In conclusion, we have successfully used a system of equations to solve for the cost of tuna and vegetarian sandwiches. By defining variables, formulating equations, and employing the elimination method, we were able to determine that a tuna sandwich costs approximately $4.85 and a vegetarian sandwich costs approximately $5.86. This problem illustrates the practical application of algebra in everyday scenarios, highlighting the importance of understanding and utilizing mathematical tools for problem-solving. Systems of equations are a fundamental concept in mathematics with wide-ranging applications. They provide a framework for representing relationships between multiple variables and solving for unknowns, making them indispensable in various fields such as science, engineering, economics, and computer science. The ability to translate real-world problems into mathematical models and solve them using algebraic techniques is a valuable skill that can be applied to a wide range of situations. This sandwich problem, while seemingly simple, encapsulates the core principles of problem-solving using systems of equations. By breaking down the problem into smaller steps, such as defining variables, formulating equations, and choosing a solution method, we were able to arrive at a precise and meaningful solution. This approach not only provides the answer but also enhances our understanding of the underlying relationships and the power of mathematical reasoning. So, as we conclude this exploration, we recognize the significance of systems of equations as a versatile and powerful tool for solving problems and making sense of the world around us.