Augmented Matrix Size For Popcorn Sales Linear System
In the realm of mathematics, linear systems serve as a powerful tool for modeling and solving real-world problems. Our specific problem revolves around deciphering popcorn sales at a bustling movie theater. We are presented with a system of two linear equations that capture the sales of large (L) and small (S) popcorn servings over two consecutive days. The equations are:
\begin{array}{l} 42L + 61S = $393 \\ 59L + 78S = $529 \end{array}
Our mission is to determine the size of the augmented matrix that represents this system. To embark on this journey, we must first understand the fundamental concepts of linear systems, matrices, and augmented matrices.
At its core, a linear system is a collection of two or more linear equations that share the same variables. Each linear equation represents a straight line when graphed, and the solution to the system is the point (or points) where all the lines intersect. In our popcorn scenario, the variables L and S represent the number of large and small popcorn servings sold, respectively. The equations themselves embody the relationship between these variables and the total sales revenue for each day. Linear systems are crucial in various fields, including engineering, economics, and computer science, because they provide a framework for modeling and solving problems involving multiple variables and constraints. The beauty of linear systems lies in their ability to simplify complex situations into manageable mathematical representations.
A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. Matrices serve as a compact and organized way to represent data and mathematical relationships. In the context of linear systems, matrices provide a convenient way to represent the coefficients and constants of the equations. Matrices are the workhorses of linear algebra, enabling us to perform operations such as addition, subtraction, multiplication, and inversion, all of which are essential for solving linear systems. The dimensions of a matrix are defined by the number of rows and columns it contains. For instance, a matrix with m rows and n columns is referred to as an m x n matrix. The elements within a matrix are identified by their row and column indices, allowing for precise referencing and manipulation of the data.
An augmented matrix is a special type of matrix that combines the coefficient matrix and the constant terms of a linear system. It provides a comprehensive representation of the system, making it easier to perform row operations and solve for the variables. The augmented matrix is formed by appending the column of constant terms to the right of the coefficient matrix, separated by a vertical line. This line serves as a visual reminder that the last column represents the constants, while the other columns represent the coefficients of the variables. The augmented matrix captures all the essential information needed to solve the linear system, streamlining the solution process and reducing the risk of errors.
Now, let's return to our popcorn sales problem. We have the following system of linear equations:
\begin{array}{l} 42L + 61S = $393 \\ 59L + 78S = $529 \end{array}
To construct the augmented matrix, we first identify the coefficients of the variables and the constant terms. The coefficients of L are 42 and 59, the coefficients of S are 61 and 78, and the constant terms are $393 and $529. We then arrange these numbers in a matrix format, with the coefficients of L in the first column, the coefficients of S in the second column, and the constant terms in the third column, separated by a vertical line. This gives us the following augmented matrix:
To determine the size of the augmented matrix, we simply count the number of rows and columns. In this case, the matrix has two rows and three columns (including the column of constant terms). Therefore, the size of the augmented matrix is 2 x 3. The size of the augmented matrix is crucial because it dictates the dimensions of the matrix and the number of operations required to solve the linear system.
The size of the augmented matrix is not just a numerical curiosity; it has significant implications for solving the linear system. The number of rows in the matrix corresponds to the number of equations in the system, while the number of columns corresponds to the number of variables plus one (for the constant terms). In our case, the 2 x 3 augmented matrix reflects the fact that we have two equations and two variables (L and S). The size of the augmented matrix also influences the choice of solution methods. For instance, Gaussian elimination, a common technique for solving linear systems, involves performing row operations on the augmented matrix to transform it into a simpler form. The number of row operations required depends on the size of the matrix.
Augmented matrices are indispensable tools for solving linear systems. They provide a structured way to represent the system and facilitate the application of various solution techniques. One of the most widely used techniques is Gaussian elimination, which involves performing elementary row operations on the augmented matrix to transform it into row-echelon form or reduced row-echelon form. These forms make it easy to identify the solutions to the system. Augmented matrices also play a crucial role in determining the consistency and uniqueness of solutions to linear systems. By analyzing the row-echelon form of the augmented matrix, we can determine whether the system has a unique solution, infinitely many solutions, or no solution at all. This information is invaluable in various applications, such as determining the feasibility of a set of constraints or the stability of a system.
In conclusion, the augmented matrix for the given linear system representing popcorn sales at a movie theater is a 2 x 3 matrix. This matrix encapsulates the essential information of the system, providing a foundation for solving for the number of large and small popcorn servings sold. The augmented matrix serves as a powerful tool for solving linear systems, enabling us to organize the coefficients and constants in a structured format and apply various solution techniques. Understanding the concept of augmented matrices is crucial for anyone working with linear systems, as it unlocks the door to solving a wide range of problems in mathematics, science, and engineering. By mastering the construction and manipulation of augmented matrices, we can confidently tackle linear systems and extract valuable insights from complex data.
- Linear System
- Augmented Matrix
- Popcorn Sales
- Matrix Size
- Gaussian Elimination
