Decoding Gillian's Book Haul Solving A System Of Equations

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Introduction

In the realm of mathematical puzzles, word problems often present intriguing scenarios that require us to unravel hidden relationships and find solutions. This article delves into a classic problem-solving exercise where we explore the intricacies of linear equations and systems of equations. Our protagonist, Gillian, embarks on a book-buying spree at a library book sale, purchasing a mix of hardcover and paperback books. The challenge lies in deciphering the exact number of each type of book Gillian acquired, given the total number of books purchased, the individual prices, and the total amount spent. To conquer this mathematical challenge, we'll employ the power of algebraic equations, setting up a system of equations that accurately represents the given information. By systematically solving this system, we'll reveal the solution and uncover the number of hardcover and paperback books Gillian added to her collection. Let's embark on this mathematical adventure and unveil the solution to Gillian's literary haul.

Problem Statement

The mathematical puzzle unfolds with Gillian's visit to a library book sale, where she indulges in her love for reading by purchasing a total of 25 books. The books are available in two formats: hardcover and paperback, each with a distinct price tag. Hardcover books are priced at $1.50 each, while paperback books are a more economical option at $0.50 each. Gillian's total expenditure at the book sale amounts to $26.50. Our mission is to determine the precise number of hardcover and paperback books that Gillian acquired. To aid us in this quest, we're provided with a system of equations that mathematically represents the scenario:

  • h + p = 25
    1. 50h + 0.50p = 26.50

Where:

  • 'h' represents the number of hardcover books
  • 'p' represents the number of paperback books

This system of equations serves as a roadmap, guiding us towards the solution. The first equation captures the total number of books purchased, while the second equation reflects the total expenditure based on the prices of each type of book. By skillfully manipulating these equations, we can isolate the values of 'h' and 'p', revealing the composition of Gillian's book collection. The challenge now lies in selecting the most efficient method to solve this system and unveil the answer. Let's explore the various techniques at our disposal and embark on the path to mathematical enlightenment.

Setting Up the Equations

To effectively tackle this problem, the crucial first step involves translating the given information into a set of mathematical equations. This process allows us to represent the relationships between the known and unknown quantities in a concise and manageable form. In this scenario, we have two primary pieces of information:

  1. The total number of books Gillian purchased
  2. The total amount Gillian spent

Let's define our variables:

  • Let 'h' represent the number of hardcover books Gillian bought.
  • Let 'p' represent the number of paperback books Gillian bought.

Now, we can translate the information into equations:

  • Equation 1 (Total number of books): Gillian purchased a total of 25 books, which means the sum of hardcover and paperback books must equal 25. This translates to the equation:
    • h + p = 25
  • Equation 2 (Total amount spent): Each hardcover book costs $1.50, and each paperback book costs $0.50. Gillian spent a total of $26.50. This translates to the equation:
      1. 50h + 0.50p = 26.50

With these two equations in hand, we have a system of equations that accurately represents the problem. The next step is to choose a method to solve this system and find the values of 'h' and 'p'. This system of equations forms the foundation for our solution, providing a clear and structured representation of the problem's constraints. By carefully crafting these equations, we've laid the groundwork for a successful mathematical journey.

Solving the System of Equations

Now that we have our system of equations established, the next step is to solve for the unknown variables, 'h' and 'p'. There are several methods we can employ to achieve this, including substitution, elimination, and matrix methods. For this particular problem, the substitution or elimination method are the most straightforward approaches. Let's demonstrate the solution using the substitution method:

  1. Solve Equation 1 for one variable:
    • We can solve the first equation (h + p = 25) for 'h':
      • h = 25 - p
  2. Substitute into Equation 2:
    • Substitute this expression for 'h' into the second equation (1.50h + 0.50p = 26.50):
        1. 50(25 - p) + 0.50p = 26.50
  3. Simplify and solve for 'p':
    • Distribute the 1.50:
      • 37.50 - 1.50p + 0.50p = 26.50
    • Combine like terms:
      • 37.50 - p = 26.50
    • Subtract 37.50 from both sides:
      • -p = -11
    • Divide both sides by -1:
      • p = 11
  4. Substitute the value of 'p' back into Equation 1 to solve for 'h':
    • h + 11 = 25
    • Subtract 11 from both sides:
      • h = 14

Therefore, the solution to the system of equations is h = 14 and p = 11. This means Gillian purchased 14 hardcover books and 11 paperback books. The substitution method has allowed us to systematically unravel the equations and arrive at a definitive solution. By carefully substituting and simplifying, we've successfully determined the composition of Gillian's literary acquisitions.

Solution and Interpretation

Having successfully navigated the algebraic terrain, we arrive at the solution: Gillian purchased 14 hardcover books and 11 paperback books. This solution provides a concrete answer to our initial question, quantifying the number of each type of book Gillian acquired at the library book sale. To ensure the accuracy of our solution, we can perform a simple verification by plugging the values of 'h' and 'p' back into the original equations:

  • Equation 1: h + p = 25
    • 14 + 11 = 25 (This holds true)
  • Equation 2: 1.50h + 0.50p = 26.50
      1. 50(14) + 0.50(11) = 21 + 5.50 = 26.50 (This also holds true)

The verification process confirms that our solution satisfies both equations, reinforcing our confidence in the accuracy of our findings. The interpretation of the solution reveals that Gillian favored hardcover books slightly more than paperback books, purchasing 14 hardcovers compared to 11 paperbacks. This could be attributed to various factors, such as Gillian's preference for the durability of hardcover books or the availability of specific titles in each format. The solution not only provides numerical answers but also offers insights into Gillian's book-buying choices. By deciphering the mathematical relationships, we gain a deeper understanding of the scenario and the underlying patterns of Gillian's literary preferences. The successful solution highlights the power of algebraic techniques in unraveling real-world problems and extracting meaningful information.

Conclusion

In this mathematical exploration, we successfully deciphered the mystery of Gillian's book purchases at the library book sale. By translating the given information into a system of equations and employing the substitution method, we determined that Gillian acquired 14 hardcover books and 11 paperback books. This exercise demonstrates the practical application of algebraic principles in solving real-world problems. The ability to set up and solve systems of equations is a valuable skill in various fields, from finance and economics to engineering and science. The problem-solving process involved several key steps:

  1. Careful reading and understanding of the problem statement: This initial step is crucial for identifying the knowns, unknowns, and the relationships between them.
  2. Defining variables: Assigning variables to represent the unknown quantities allows us to express the problem mathematically.
  3. Formulating equations: Translating the given information into equations is the core of the problem-solving process, capturing the essential relationships between the variables.
  4. Solving the system of equations: Choosing an appropriate method (substitution, elimination, etc.) and systematically applying it to find the values of the unknowns.
  5. Interpreting the solution: Understanding the meaning of the solution in the context of the original problem and verifying its accuracy.

By mastering these steps, we can confidently tackle a wide range of mathematical challenges. The story of Gillian's book purchases serves as a compelling example of how mathematical tools can be used to unravel everyday mysteries and make informed decisions. The power of algebra lies in its ability to transform complex scenarios into manageable equations, providing a framework for logical reasoning and problem-solving. As we continue our mathematical journey, we'll encounter many more intriguing problems that can be solved using the same fundamental principles.