Problem 8 - A Graphical Approach to Precalculus with Limits

Question

Write the augmented matrix for each system. Do not solve the system. $$\begin{aligned} &3 x+5 y=-13\\\ &2 x+3 y=-9 \end{aligned}$$

Step-by-Step Solution

Answer

The augmented matrix is \[ \begin{bmatrix} 3 & 5 & | & -13 \\ 2 & 3 & | & -9 \end{bmatrix} \].

Step 1: Identify Coefficients and Constants

To write the augmented matrix, first identify the coefficients of the variables and the constants from the given system of equations: \[3x + 5y = -13\] and \[2x + 3y = -9\]. The coefficients for the first equation are 3 and 5, and the constant is -13. For the second equation, the coefficients are 2 and 3, and the constant is -9.

Step 2: Assemble the Coefficient Matrix

Create the coefficient matrix by arranging the coefficients of the variables into a matrix format. For the given system of equations, the coefficient matrix is: \[ \begin{bmatrix} 3 & 5 \ 2 & 3 \end{bmatrix} \].

Step 3: Form the Augmented Matrix

The augmented matrix is formed by appending the constants from each equation as an additional column to the coefficient matrix. Use the constants identified in Step 1 to form the augmented matrix: \[ \begin{bmatrix} 3 & 5 & | & -13 \ 2 & 3 & | & -9 \end{bmatrix} \]. Here, the vertical bar sometimes indicates the separation between the coefficient matrix and the column of constants.

Key Concepts

Coefficient Matrix

When working with a system of equations, the coefficient matrix plays a vital role. It's essentially a neat way to organize the coefficients of the variables in a system into a matrix format.

In our original exercise, we have the equations:\[3x + 5y = -13\]and\[2x + 3y = -9\].

The respective coefficients of variables \(x\) and \(y\) in the first equation are 3 and 5. In the second equation, they are 2 and 3. These coefficients are then placed into separate rows to form the coefficient matrix:

• The first row represents the coefficients from the first equation: 3 and 5.
• The second row contains the coefficients from the second equation: 2 and 3.

Thus, the coefficient matrix is:\[\begin{bmatrix} 3 & 5 \2 & 3 \end{bmatrix}\] This matrix effectively captures the essential information about the relationships between the variables in a compact form.

System of Equations

A system of equations is a collection of two or more equations with a shared set of variables. In the given exercise, the system consists of the two equations:

• \(3x + 5y = -13\)
• \(2x + 3y = -9\)

The goal when dealing with such a system varies depending on the problem context; sometimes it is to find the values of the variables that satisfy all the equations simultaneously. However, in this exercise, we are only setting up the foundational step of expressing the system in a matrix format.

The concept of a system of equations is fundamental in linear algebra as it lays the groundwork for understanding how different variables interact and depend on each other. Representing a system through matrices allows for a more visual and structured approach to analyze and solve it.

Matrix Format

The matrix format is a convenient way to represent a system of linear equations. By using this format, the problem can be analyzed using linear algebra techniques, which often simplify complex operations.

For our system, transitioning to matrix format involves two crucial steps:

• Creating the coefficient matrix, which captures the coefficients of the variables.
• Forming the augmented matrix by appending the constant terms from the equations as an additional column.

This results in an augmented matrix:\[\begin{bmatrix} 3 & 5 & | & -13 \2 & 3 & | & -9 \end{bmatrix}\] The vertical bar often used in augmented matrices helps to distinguish between the coefficient matrix and the constants. This design prepares you for further manipulations, such as performing row operations when solving the system. Understanding matrix format not only aids in organizing information but also sets the stage for efficient computation methods in solving linear equations.