Solving For Fruit Prices Using Systems Of Equations

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Introduction

In this mathematical exploration, we delve into a real-world scenario involving the purchase of fruits, specifically apples and bananas. The problem presents a classic system of equations that allows us to determine the individual costs per pound of each fruit. Lydia's and Ari's purchases provide us with the necessary information to set up these equations and solve for the unknowns. This article aims to dissect the problem, understand the underlying mathematical principles, and provide a comprehensive solution. By the end, readers will gain a deeper understanding of how systems of equations can be applied to everyday situations, enhancing their problem-solving skills and mathematical intuition.

Setting up the Equations

To begin, let's define our variables. Let x represent the cost per pound of apples and y represent the cost per pound of bananas. According to the problem, Lydia buys 5 pounds of apples and 3 pounds of bananas for a total of $8.50. This can be translated into the following equation:

5x + 3y = 8.50

Similarly, Ari buys 3 pounds of apples and 2 pounds of bananas for a total of $5.25. This gives us the second equation:

3x + 2y = 5.25

These two equations form a system of linear equations. Solving this system will reveal the values of x and y, which represent the cost per pound of apples and bananas, respectively. The system of equations is a fundamental concept in algebra, allowing us to solve for multiple unknowns when we have a corresponding number of independent equations. In this case, we have two unknowns (the cost of apples and bananas) and two equations, making it a solvable system. The beauty of this approach lies in its ability to model real-world scenarios with mathematical precision, providing a powerful tool for problem-solving.

Solving the System of Equations

Now that we have our system of equations, we can explore different methods to find the solution. There are several techniques available, including substitution, elimination, and graphical methods. For this problem, we will demonstrate the elimination method, which is particularly efficient when dealing with linear equations. The elimination method involves manipulating the equations so that when they are added or subtracted, one of the variables is eliminated, leaving us with a single equation in one variable. Let's proceed step-by-step to solve our system.

Elimination Method

Our system of equations is:

  1. 5x + 3y = 8.50
  2. 3x + 2y = 5.25

To eliminate one of the variables, we need to make the coefficients of either x or y the same (but with opposite signs) in both equations. Let's eliminate x. To do this, we'll multiply the first equation by 3 and the second equation by -5:

  1. 3 * (5x + 3y) = 3 * 8.50 => 15x + 9y = 25.50
  2. -5 * (3x + 2y) = -5 * 5.25 => -15x - 10y = -26.25

Now, we add the two modified equations:

(15x + 9y) + (-15x - 10y) = 25.50 + (-26.25)

This simplifies to:

-y = -0.75

Dividing both sides by -1, we find:

y = 0.75

Now that we have the value of y, we can substitute it back into one of the original equations to find x. Let's use the first original equation:

5x + 3(0.75) = 8.50

Simplifying:

5x + 2.25 = 8.50

Subtracting 2.25 from both sides:

5x = 6.25

Dividing both sides by 5:

x = 1.25

Thus, we have found that x = 1.25 and y = 0.75.

Solution and Interpretation

The solution to our system of equations is x = 1.25 and y = 0.75. This means that the cost per pound of apples is $1.25, and the cost per pound of bananas is $0.75. This is a clear and concise answer to our original problem. The process of setting up and solving the system of equations allowed us to break down a real-world scenario into manageable mathematical components. This approach is not only applicable to fruit purchases but can be extended to a wide range of situations, from calculating the costs of different items to determining the optimal mix of ingredients in a recipe.

Real-World Applications and Importance of Systems of Equations

The application of systems of equations extends far beyond simple scenarios like buying fruits. In the real world, systems of equations are used extensively in various fields, including:

  • Economics: Economists use systems of equations to model and predict economic trends, such as supply and demand, inflation, and unemployment rates. These models help policymakers make informed decisions about fiscal and monetary policy.
  • Engineering: Engineers use systems of equations to design structures, circuits, and systems. For example, structural engineers use equations to calculate the forces acting on a bridge or building, ensuring its stability and safety. Electrical engineers use systems of equations to analyze circuits and design electronic devices.
  • Computer Science: In computer graphics and simulations, systems of equations are used to model the behavior of objects and environments. This allows for the creation of realistic and interactive experiences in video games, virtual reality, and other applications.
  • Finance: Financial analysts use systems of equations to model investment portfolios, assess risk, and make predictions about market behavior. These models help investors make informed decisions about where to allocate their capital.
  • Science: Scientists in various fields, such as physics, chemistry, and biology, use systems of equations to model natural phenomena. For example, physicists use equations to describe the motion of particles, chemists use equations to model chemical reactions, and biologists use equations to model population dynamics.

Further Exploration and Practice

To further enhance your understanding of systems of equations, consider exploring additional problems and scenarios. Practice solving systems of equations using different methods, such as substitution and graphing, to develop a well-rounded skill set. You can also investigate real-world applications of systems of equations in areas that interest you, such as economics, engineering, or computer science. By continuously practicing and exploring, you will deepen your understanding and appreciation for the power and versatility of this fundamental mathematical concept.

Conclusion

In conclusion, the problem of determining the cost of apples and bananas using a system of equations demonstrates the practical application of mathematics in everyday life. By setting up and solving the equations, we were able to find the individual costs per pound of each fruit. This process not only provides a solution to the specific problem but also illustrates the broader importance of systems of equations in various fields. From economics to engineering, systems of equations are a powerful tool for modeling and solving complex problems. By understanding and mastering this concept, we can enhance our problem-solving abilities and gain a deeper appreciation for the role of mathematics in the world around us. Remember, the key to success in mathematics is practice and exploration. Keep challenging yourself with new problems and scenarios, and you will continue to grow your skills and knowledge.

By understanding how to set up and solve such systems, we not only address the immediate question but also equip ourselves with a valuable tool applicable in diverse real-world situations. The ability to translate word problems into mathematical equations and find solutions is a critical skill in mathematics and beyond, fostering logical thinking and analytical capabilities.