Solving A System Of Equations To Find The Price Of A Pencil
Introduction
In this article, we will delve into a mathematical problem involving a system of equations. System of equations are fundamental in various fields, including mathematics, physics, and economics, as they allow us to model and solve real-world problems with multiple variables and constraints. This problem, adapted from UNIFESP-04, presents a scenario involving the prices of pencils and a pencil case in a bookstore. Our goal is to determine the price of a single pencil by solving the system of equations derived from the given information. This exercise will not only enhance our understanding of solving systems of equations but also demonstrate their practical application in everyday situations. Understanding how to solve this system of equations is crucial for anyone looking to improve their mathematical skills and problem-solving abilities. We will break down the problem step by step, making it easy to follow and understand the logic behind each step. By the end of this article, you will have a clear understanding of how to approach and solve similar problems. The ability to solve such problems is not just about getting the right answer; it’s about developing a logical and analytical mindset that can be applied to various aspects of life. So, let’s dive in and explore the world of systems of equations and see how they can help us solve real-world problems.
Problem Statement
The problem states that in a certain bookstore, the sum of the purchase prices of two pencils and a pencil case is R$10.00. Additionally, the price of the pencil case is R$5.00 cheaper than the price of three pencils. This information leads to the system of equations: 2x + y = 10 and y = 3x - 5, where 'x' represents the price of a pencil and 'y' represents the price of the pencil case. Our task is to determine the price of acquiring one pencil. This is a classic problem that requires us to use our knowledge of algebra and systems of equations. The problem is designed to test our ability to translate word problems into mathematical equations and then solve them. It also highlights the importance of understanding the relationships between different variables in a given situation. Before we jump into the solution, let's take a moment to understand the key components of the problem. We have two variables, 'x' and 'y', and two equations that relate these variables. This means we have a well-defined system of equations that can be solved using various methods, such as substitution or elimination. Understanding the problem statement is the first step towards finding the correct solution. Once we have a clear understanding of what the problem is asking, we can then proceed to develop a strategy to solve it. This involves identifying the key information, translating it into mathematical equations, and then applying the appropriate techniques to solve the system of equations.
Setting up the Equations
To solve this problem, we first need to translate the given information into mathematical equations. Let's denote the price of one pencil as 'x' and the price of the pencil case as 'y'.
The first piece of information tells us that the sum of the prices of two pencils and one pencil case is R$10.00. This can be written as the equation:
2x + y = 10
The second piece of information states that the price of the pencil case is R$5.00 cheaper than the price of three pencils. This can be written as the equation:
y = 3x - 5
Now we have a system of two equations with two variables:
- 2x + y = 10
- y = 3x - 5
This system of equations represents the relationships between the prices of pencils and the pencil case. Setting up these equations correctly is crucial for solving the problem. If the equations are not set up correctly, the solution will be incorrect. The process of translating word problems into mathematical equations is a fundamental skill in algebra. It requires careful reading and understanding of the problem statement, identifying the key information, and then expressing that information in mathematical form. In this case, we have successfully translated the given information into a system of equations that we can now solve. The next step is to choose a method for solving the system of equations. There are several methods we can use, such as substitution, elimination, or graphing. In this case, the substitution method seems to be the most straightforward, as we already have one equation solved for 'y'. We can substitute the expression for 'y' from the second equation into the first equation, which will give us an equation with only one variable, 'x'. This will allow us to solve for 'x', which is the price of a pencil.
Solving the System of Equations
We can use the substitution method to solve this system. Since the second equation is already solved for 'y', we can substitute the expression '3x - 5' for 'y' in the first equation:
2x + (3x - 5) = 10
Now, we simplify the equation:
2x + 3x - 5 = 10
5x - 5 = 10
Next, we add 5 to both sides of the equation:
5x = 15
Finally, we divide both sides by 5 to solve for 'x':
x = 3
So, the price of one pencil is R$3.00. Now that we have found the value of 'x', we can also find the value of 'y' by substituting x = 3 into the second equation:
y = 3(3) - 5
y = 9 - 5
y = 4
Therefore, the price of the pencil case is R$4.00. However, the question asks for the price of one pencil, which we have already found to be R$3.00. Solving the system of equations involves a series of algebraic manipulations that require a clear understanding of the rules of algebra. Each step must be performed carefully to avoid errors. The substitution method is a powerful technique for solving systems of equations, especially when one equation is already solved for one of the variables. It allows us to reduce the system of equations to a single equation with one variable, which is much easier to solve. Once we have found the value of one variable, we can then substitute it back into one of the original equations to find the value of the other variable. This process ensures that we have a complete solution to the system of equations.
Finding the Price of a Pencil
From the solution of the system of equations, we found that x = 3. Since 'x' represents the price of one pencil, the price of a pencil is R$3.00. This result answers the question posed in the problem statement. The process of finding the price of a pencil involved several steps, from setting up the equations to solving them. Each step required a clear understanding of the problem and the mathematical techniques needed to solve it. The final answer is a direct result of the algebraic manipulations we performed. It is important to note that the answer should always be checked against the original problem statement to ensure that it makes sense in the context of the problem. In this case, the price of R$3.00 for a pencil seems reasonable, given the other information provided in the problem. The ability to solve problems like this is a valuable skill that can be applied in various situations. It demonstrates a strong understanding of algebra and problem-solving techniques. It also highlights the importance of being able to translate real-world scenarios into mathematical models and then solve them using appropriate methods. This is a skill that is highly valued in many fields, including science, engineering, and finance. So, mastering these techniques is an investment in your future.
Conclusion
In conclusion, the price of acquiring one pencil is R$3.00. We arrived at this solution by setting up a system of equations based on the given information and then solving it using the substitution method. This problem demonstrates the practical application of systems of equations in solving real-world scenarios. The process involved translating the word problem into mathematical equations, solving the equations using algebraic techniques, and then interpreting the solution in the context of the problem. This is a common approach to solving mathematical problems in various fields. The ability to solve systems of equations is a valuable skill that can be applied in many different contexts. It requires a clear understanding of algebraic principles and problem-solving strategies. By working through problems like this, we can improve our mathematical skills and develop a logical and analytical mindset. The solution to this problem also highlights the importance of careful reading and understanding of the problem statement. It is crucial to identify the key information and then translate it into mathematical form accurately. Any errors in this process can lead to an incorrect solution. Therefore, it is important to practice these skills and develop a systematic approach to solving mathematical problems. This will not only help you succeed in your math classes but also prepare you for the challenges you will face in your future career. Mathematical problem-solving is a fundamental skill that is valued in many different professions. So, mastering these techniques is an investment in your future success.
