Problem 14 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the augmented matrix for the system of linear equations. What is the dimension of the augmented matrix? $$\left\\{\begin{array}{l} 7 x+4 y=22 \\ 5 x-9 y=15 \end{array}\right.$$

Step-by-Step Solution

Answer

The augmented matrix for the system of equations is: \[ \begin{{array}}{{cc|c}}7 & 4 & 22 \5 & -9 & 15 \\end{{array}} \] and its dimension is 2x3.

Step 1: Express the linear equations in the form Ax+By=C

For the system of equations, we have two equations expressed in the form Ax+By=C. We have \(7x + 4y = 22\) and \(5x - 9y = 15\).

Step 2: Arrange the coefficients in a matrix

The augmented matrix will consist of the coefficients separated by a line to indicate the equality. Each equation forms a row and the coefficients of like terms form the columns. The first row of the matrix would consist of the coefficients of the first equation. So the first row for the augmented matrix is \( [7, 4 | 22] \). For the second row of the matrix, do the same for the second equation. The second row for the augmented matrix is \( [5, -9 | 15] \).

Step 3: Write the full augmented matrix and determine its dimension

Putting the two rows together, the augmented matrix is: \[ \begin{{array}}{{cc|c}}7 & 4 & 22 \5 & -9 & 15 \\end{{array}} \] The dimension of a matrix is given as 'number of rows' x 'number of columns'. This matrix has 2 rows and 3 columns so its dimension is 2x3.

Key Concepts

Understanding Systems of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. The objective is to find values for the variables that satisfy all equations in the system simultaneously. In the exercise, there are two linear equations involving "x" and "y":

• Equation 1: \(7x + 4y = 22\)
• Equation 2: \(5x - 9y = 15\)

Finding solutions to these equations means finding values of "x" and "y" that make both equations true at the same time. In this context, using matrices offers an efficient way to represent and solve these sets of equations.

Grasping Matrix Dimensions

When dealing with matrices, understanding matrix dimensions is crucial. The dimension of a matrix tells us how many rows and columns it has. For any matrix, the format for dimensions is typically noted as "number of rows" \( \times \) "number of columns".
In the solution exercise, the augmented matrix formed was \(\begin{array}{cc|c}7 & 4 & 22 \5 & -9 & 15 \\end{array}\).
The matrix has:

• 2 rows
• 3 columns

These dimensions help us understand the organization of the matrix, as they influence how we manipulate and solve it.

Matrix Representation of Equations

A matrix is a rectangular array of numbers. In the context of systems of linear equations, they are used to organize information systematically. The augmented matrix is particularly useful because it includes the coefficients of the variables as well as the constants from the equations.
For example, a system of equations such as those in the exercise can be transformed into an augmented matrix, represented as:

• First row corresponds to the first equation's coefficients: \([7, 4 | 22]\)
• Second row corresponds to the second equation's coefficients: \([5, -9 | 15]\)

The vertical line "|" within the matrix separates the coefficients of the variables from the constants. This distinct separation helps when performing operations to solve the system of equations, such as row reduction, making it easier to focus on both parts independently.