Solving Gillian's Book Purchase A System Of Equations Example

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Introduction: Decoding the Bookworm's Budget

In the realm of mathematical puzzles, we often encounter scenarios that require us to dissect information, formulate equations, and arrive at logical conclusions. This article delves into one such scenario, a captivating tale of Gillian, a book enthusiast who embarks on a literary shopping spree at a library book sale. Our mission is to unravel the financial intricacies of her book purchase, exploring the interplay between the number of books, their respective costs, and the total amount spent. This problem is a classic example of a system of equations, a fundamental concept in algebra that allows us to solve for multiple unknowns using multiple equations. By carefully analyzing the given information and employing algebraic techniques, we can determine the precise number of hardcover and paperback books Gillian acquired.

Understanding the problem is the first step towards solving it. We know that Gillian purchased a total of 25 books, a mix of hardcovers and paperbacks. Each hardcover book came at a price of $1.50, while each paperback cost $0.50. Gillian's total expenditure amounted to $26.50. Our objective is to determine how many of each type of book she bought. To achieve this, we'll translate the word problem into a system of two equations with two unknowns. Let's denote the number of hardcover books as 'h' and the number of paperback books as 'p'. We can then formulate our equations based on the information provided:

  • The total number of books: h + p = 25
  • The total cost of the books: 1.50h + 0.50p = 26.50

With these equations in hand, we can now employ various algebraic methods to solve for 'h' and 'p'. This problem not only tests our mathematical prowess but also highlights the practical applications of algebra in everyday scenarios. Whether it's managing personal finances, planning a budget, or simply satisfying our curiosity about a bookworm's spending habits, the ability to solve systems of equations proves to be an invaluable skill. So, let's embark on this literary-mathematical journey and decipher the cost of Gillian's book haul.

Setting Up the Equations: Translating Words into Algebra

To effectively solve any word problem, the crucial first step lies in transforming the given information into a mathematical framework. In Gillian's case, we need to translate the narrative about her book purchases into a system of equations that accurately represents the relationships between the quantities involved. This process involves identifying the unknowns, assigning variables to them, and then formulating equations based on the provided facts.

As we established in the introduction, our unknowns are the number of hardcover books (which we'll denote as 'h') and the number of paperback books (denoted as 'p'). Now, let's carefully examine the information provided in the problem statement and construct our equations.

The first piece of information is that Gillian purchased a total of 25 books. This translates directly into our first equation:

h + p = 25

This equation simply states that the sum of the number of hardcover books and the number of paperback books is equal to 25. It's a straightforward representation of the total quantity of books purchased.

The second piece of information pertains to the cost of the books. We know that each hardcover book costs $1.50, and each paperback book costs $0.50. Gillian spent a total of $26.50. This information allows us to formulate our second equation:

1.  50h + 0.50p = 26.50

This equation represents the total cost of the books. The term '1.50h' represents the total cost of the hardcover books (price per hardcover multiplied by the number of hardcovers), and the term '0.50p' represents the total cost of the paperback books. The sum of these two terms equals Gillian's total expenditure of $26.50.

Now that we have successfully translated the word problem into a system of two equations, we have a solid foundation for solving for our unknowns. The next step involves choosing an appropriate method for solving this system, such as substitution or elimination. By manipulating these equations algebraically, we can isolate the variables and determine the values of 'h' and 'p', thus revealing the number of hardcover and paperback books Gillian purchased. The process of setting up these equations is fundamental to problem-solving in mathematics, as it allows us to represent real-world scenarios in a concise and manageable form.

Solving the System: Unveiling the Number of Books

With our system of equations neatly established, the next step is to embark on the mathematical journey of solving it. We have two equations and two unknowns, which means we have a solvable system. There are several methods we can employ to find the values of 'h' (the number of hardcover books) and 'p' (the number of paperback books). Two common methods are substitution and elimination. Let's explore the substitution method in this case.

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This effectively reduces the system to a single equation with a single unknown, which is much easier to solve. Let's start with our first equation:

h + p = 25

We can easily solve this equation for 'h' by subtracting 'p' from both sides:

h = 25 - p

Now we have an expression for 'h' in terms of 'p'. We can substitute this expression into our second equation:

1.  50h + 0.50p = 26.50

Replacing 'h' with '(25 - p)', we get:

1.  50(25 - p) + 0.50p = 26.50

Now we have a single equation with only one unknown, 'p'. Let's simplify and solve for 'p':

3.  5 - 1.50p + 0.50p = 26.50
3.  5 - p = 26.50
-p = 26.50 - 37.5
-p = -11
p = 11

So, we've found that 'p' (the number of paperback books) is 11. Now that we know 'p', we can easily find 'h' using the expression we derived earlier:

h = 25 - p
h = 25 - 11
h = 14

Therefore, 'h' (the number of hardcover books) is 14. We have successfully solved the system of equations! Gillian purchased 14 hardcover books and 11 paperback books. This meticulous process of algebraic manipulation demonstrates the power of equations in unraveling real-world scenarios.

Verifying the Solution: Ensuring Accuracy in Our Calculations

In the realm of mathematics, precision and accuracy are paramount. After solving a problem, it's crucial to verify the solution to ensure that it aligns with the initial conditions and constraints. This step helps us catch any potential errors in our calculations and provides confidence in our final answer. In the case of Gillian's book purchase, we can verify our solution by plugging the values we found for 'h' and 'p' back into our original equations.

We determined that Gillian purchased 14 hardcover books (h = 14) and 11 paperback books (p = 11). Let's revisit our original system of equations:

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

First, let's plug our values into the first equation:

14 + 11 = 25

This equation holds true, as 14 + 11 indeed equals 25. This confirms that our solution satisfies the condition that Gillian purchased a total of 25 books.

Next, let's plug our values into the second equation:

1.  50(14) + 0.50(11) = 26.50
2.  00 + 5.50 = 26.50
3.  50 = 26.50

This equation also holds true, as the total cost calculated using our values matches the given total expenditure of $26.50. This further validates our solution.

By successfully verifying our solution in both equations, we can confidently conclude that our calculations are accurate. Gillian indeed purchased 14 hardcover books and 11 paperback books. This step of validation is an integral part of the problem-solving process, ensuring the reliability and correctness of our results.

Conclusion: The Power of Systems of Equations

In this article, we embarked on a mathematical exploration of Gillian's book purchase at a library book sale. We successfully unraveled the cost of her book haul by employing the concept of systems of equations, a fundamental tool in algebra. By translating the word problem into a mathematical framework, we were able to determine the precise number of hardcover and paperback books Gillian acquired.

We began by carefully analyzing the given information and identifying the unknowns: the number of hardcover books (h) and the number of paperback books (p). We then formulated two equations based on the provided facts:

  • The total number of books: h + p = 25
  • The total cost of the books: 1.50h + 0.50p = 26.50

With our system of equations established, we employed the substitution method to solve for 'h' and 'p'. We found that Gillian purchased 14 hardcover books and 11 paperback books. To ensure the accuracy of our solution, we verified our results by plugging the values back into our original equations.

This problem serves as a compelling illustration of the power and versatility of systems of equations. They provide a structured approach to solving problems involving multiple unknowns and relationships. Whether it's managing personal finances, planning a budget, or deciphering a bookworm's spending habits, the ability to formulate and solve systems of equations proves to be an invaluable skill. The process of translating real-world scenarios into mathematical equations allows us to analyze complex situations, make informed decisions, and arrive at logical conclusions.

In conclusion, Gillian's library book haul provided us with an engaging context for exploring the concept of systems of equations. By mastering this mathematical tool, we empower ourselves to tackle a wide range of problems and gain a deeper understanding of the world around us. The journey from word problem to algebraic solution highlights the elegance and practicality of mathematics in everyday life.