Solving Systems Of Equations Textbook And Workbook Problem

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Introduction

In the realm of mathematics, we often encounter problems that require us to apply our knowledge of equations and systems of equations to real-world scenarios. These problems not only help us solidify our understanding of mathematical concepts but also enhance our problem-solving skills, which are crucial in various aspects of life. This article delves into a classic example of such a problem, involving a mix of textbooks and workbooks packed in a box, each with a different weight, and challenges us to determine the exact number of each type of book. Let's explore the steps to solve this mathematical puzzle and uncover the power of algebraic techniques.

Problem Statement

Imagine Janet, a diligent student or teacher, who has packed a total of 50 textbooks and workbooks into a sturdy box. However, in her haste, she forgot the exact number of each type of book. What she does remember is that each textbook weighs 2 pounds, while each workbook is a lighter 0.5 pounds. Adding to the complexity, Janet recalls that the total weight of all the books in the box is a hefty 55 pounds. The challenge now is to figure out exactly how many textbooks and how many workbooks are nestled within the box. This problem is a perfect example of how a system of equations can be used to solve a real-world mathematical problem.

Setting Up the Equations

To solve this problem effectively, we need to translate the given information into mathematical equations. This is a fundamental step in tackling many mathematical problems. Let's use variables to represent the unknowns:

  • Let 'x' represent the number of textbooks.
  • Let 'y' represent the number of workbooks.

From the problem statement, we can derive two key pieces of information:

  1. The total number of books: Janet packed a total of 50 books, which means the sum of textbooks and workbooks is 50. This can be written as the equation:

    x + y = 50

  2. The total weight of the books: Each textbook weighs 2 pounds, and each workbook weighs 0.5 pounds. The total weight is 55 pounds. This translates to the equation:

    2x + 0. 5y = 55

Now, we have a system of two linear equations with two variables:

x + y = 50
2x + 0.5y = 55

This system of equations is the key to unlocking the solution to our problem. By solving this system, we can determine the values of 'x' and 'y', which represent the number of textbooks and workbooks, respectively. The next step is to choose a method to solve this system, and we'll explore a couple of common approaches in the following sections.

Solving the System of Equations

Now that we have our system of equations, we can use a couple of methods to find the solution.

Method 1: Substitution

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This eliminates one variable, allowing us to solve for the remaining one.

  1. Solve the first equation for x: x + y = 50 x = 50 - y
  2. Substitute this expression for x into the second equation: 2(50 - y) + 0.5y = 55
  3. Simplify and solve for y: 100 - 2y + 0.5y = 55 -1.5y = -45 y = 30
  4. Substitute the value of y back into the equation x = 50 - y to find x: x = 50 - 30 x = 20

Method 2: Elimination

The elimination method involves manipulating the equations so that the coefficients of one variable are opposites. Then, we add the equations together, eliminating that variable.

  1. Multiply the first equation by -2 to make the coefficients of x opposites: -2(x + y) = -2(50) -2x - 2y = -100
  2. Add the modified first equation to the second equation: (-2x - 2y) + (2x + 0.5y) = -100 + 55 -1.5y = -45
  3. Solve for y: y = 30
  4. Substitute the value of y back into the equation x + y = 50 to find x: x + 30 = 50 x = 20

Both methods lead us to the same solution: x = 20 and y = 30. This means there are 20 textbooks and 30 workbooks in the box. Understanding these methods is crucial for solving various mathematical problems involving multiple variables.

Verifying the Solution

Before we declare victory, it's crucial to verify our solution. This step ensures that our calculated values satisfy the conditions of the original problem. We need to plug the values of x and y back into our original equations to confirm that they hold true.

  1. Check the total number of books:

    x + y = 50

    20 + 30 = 50

    50 = 50 (The equation holds true)

  2. Check the total weight of the books:

    2x + 0. 5y = 55

    2(20) + 0.5(30) = 55

    40 + 15 = 55

    55 = 55 (The equation holds true)

Since both equations are satisfied by our values of x = 20 and y = 30, we can confidently conclude that our solution is correct. This verification step is a cornerstone of mathematical problem-solving, ensuring accuracy and preventing errors.

Conclusion

In this article, we successfully tackled a real-world mathematical problem involving a system of equations. We were able to determine that Janet packed 20 textbooks and 30 workbooks into the box. By translating the given information into mathematical equations and employing methods such as substitution and elimination, we were able to arrive at the solution. The importance of verifying the solution was also highlighted, emphasizing the need for accuracy in mathematical problem-solving.

This example showcases the power and applicability of mathematics in everyday situations. Systems of equations are not just abstract concepts; they are tools that can help us solve practical problems. As we continue our mathematical journey, we will encounter even more complex and intriguing problems, but the fundamental principles we've learned here will serve as a strong foundation. This mathematical exploration demonstrates the practical application of algebra in solving everyday puzzles.

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