Augmented Matrix Size For Linear System Representing Popcorn Sales

In the realm of linear algebra, systems of equations often serve as mathematical models for real-world scenarios. Consider a scenario where we're tracking the sales of large and small popcorn buckets at a movie theater over consecutive days. This situation can be elegantly represented using a system of linear equations. Let's delve into this concept and explore how to determine the size of the augmented matrix associated with such a system.

Understanding the Linear System

The provided linear system is:

42L + 61S = 393
59L + 78S = 529

Here, 'L' represents the number of large popcorn buckets sold, and 'S' represents the number of small popcorn buckets sold. The two equations correspond to the sales data for two consecutive days. The constants on the right-hand side (393 and 529) represent the total revenue generated from popcorn sales on those respective days.

This system of equations encapsulates a fundamental relationship between the quantities of large and small popcorn buckets sold and the resulting revenue. Our goal is to decipher the structure of this system and represent it in a matrix form, specifically as an augmented matrix.

What is an Augmented Matrix?

An augmented matrix is a powerful tool for representing and solving systems of linear equations. It's essentially a compact way of writing the coefficients and constants of the equations in a matrix format. To construct an augmented matrix, we extract the coefficients of the variables and the constants from the equations and arrange them in a rectangular array. A vertical line typically separates the coefficient matrix from the constant vector, visually distinguishing the two parts.

Consider a general system of linear equations:

a₁₁x₁ + a₁₂x₂ + ... + a₁ₙxₙ = b₁
a₂₁x₁ + a₂₂x₂ + ... + a₂ₙxₙ = b₂
...
am₁x₁ + am₂x₂ + ... + amₙxₙ = bm

The augmented matrix for this system would be:

[ a₁₁ a₁₂ ... a₁ₙ | b₁ ]
[ a₂₁ a₂₂ ... a₂ₙ | b₂ ]
[ ... ... ... ... | ... ]
[ am₁ am₂ ... amₙ | bm ]

Notice how the coefficients (aᵢⱼ) form the main part of the matrix, and the constants (bᵢ) are appended as the last column, separated by a vertical line. The augmented matrix provides a concise and organized representation of the linear system, facilitating various matrix operations for solving the system.

Constructing the Augmented Matrix for the Popcorn Sales System

Now, let's apply this concept to our popcorn sales system. We have the following equations:

42L + 61S = 393
59L + 78S = 529

To form the augmented matrix, we extract the coefficients of L and S and the constants on the right-hand side:

• Coefficients of L: 42 and 59
• Coefficients of S: 61 and 78
• Constants: 393 and 529

Arranging these values in a matrix format, we get the augmented matrix:

[ 42 61 | 393 ]
[ 59 78 | 529 ]

This matrix represents the same information as the original system of equations but in a more compact and structured form. The first two columns correspond to the coefficients of L and S, respectively, and the last column represents the constants.

Determining the Size of the Augmented Matrix

The size of a matrix is defined by its dimensions, which are the number of rows and columns. In the augmented matrix we constructed:

[ 42 61 | 393 ]
[ 59 78 | 529 ]

• Number of rows: There are two horizontal rows in the matrix.
• Number of columns: There are three vertical columns (two for the coefficients and one for the constants).

Therefore, the size of the augmented matrix is 2 rows by 3 columns, which is written as 2 × 3.

Why is the Size of the Augmented Matrix Important?

The size of the augmented matrix is a crucial characteristic that reveals important information about the system of linear equations it represents. Specifically:

1. Number of Equations: The number of rows in the augmented matrix corresponds to the number of equations in the system. In our popcorn sales example, the 2 × 3 augmented matrix indicates that we have two equations.

2. Number of Variables: The number of columns in the coefficient part of the augmented matrix (before the vertical line) corresponds to the number of variables in the system. In our example, the coefficient part has two columns, indicating that we have two variables (L and S).

3. Solvability: The relative number of rows and columns in the augmented matrix can provide insights into the solvability of the system. For instance, if the number of rows is less than the number of columns, the system may have infinitely many solutions or no solutions. If the number of rows is equal to the number of columns, the system may have a unique solution.

4. Matrix Operations: The size of the augmented matrix dictates the types of matrix operations that can be performed on it. For example, row operations, which are essential for solving systems of equations using methods like Gaussian elimination, are influenced by the matrix size.

In essence, the size of the augmented matrix serves as a fundamental descriptor of the linear system, providing valuable information about its structure and properties.

Answer to the Question

The question asks for the size of the augmented matrix for the given system of equations. As we determined, the augmented matrix is:

[ 42 61 | 393 ]
[ 59 78 | 529 ]

This matrix has 2 rows and 3 columns, so its size is 2 × 3.

Therefore, the correct answer is A. 2 × 3.

Conclusion

In this exploration, we've dissected a linear system representing popcorn sales and uncovered the significance of the augmented matrix. We learned how to construct an augmented matrix from a system of equations and, more importantly, how to determine its size. The size of the augmented matrix serves as a key indicator of the system's structure, providing insights into the number of equations, variables, and potential solvability. Understanding these concepts is fundamental to solving linear systems and applying them to real-world problems, such as modeling sales trends or analyzing financial data.