Question

Convert the given system of linear equations into an augmented matrix. $$ \begin{aligned} 3 x_{1}-2 x_{2} &=4 \\ 2 x_{1} &=3 \\ -x_{1}+9 x_{2} &=8 \\ 5 x_{1}-7 x_{2} &=13 \end{aligned} $$

Short answer

The augmented matrix is \(\begin{bmatrix} 3 & -2 & | & 4 \\ 2 & 0 & | & 3 \\ -1 & 9 & | & 8 \\ 5 & -7 & | & 13 \end{bmatrix}\).

Step-by-step solution

Write the Coefficients Matrix

Identify the coefficients of each variable in the system of equations. The coefficients are as follows: \(3, -2\); \(2, 0\); \(-1, 9\); and \(5, -7\). These will form the primary part of your augmented matrix.

Identify the Right-Hand Side Constants

Extract the constants from the right side of each equation. They are \(4, 3, 8,\) and \(13\). These will become the augmented part of the matrix.

Form the Augmented Matrix

Combine the coefficients and constants into matrix form. Your matrix will be:\[\begin{bmatrix} 3 & -2 & | & 4 \2 & 0 & | & 3 \-1 & 9 & | & 8 \5 & -7 & | & 13 \end{bmatrix}\]Ensure the vertical line separates the coefficient matrix and the constants.

Key concepts

System of Linear Equations

A system of linear equations is simply a set of two or more linear equations that have common variables. In this scenario, we are dealing with four equations that involve two variables, \(x_1\) and \(-x_2\). Each linear equation can be visualized as a line in two-dimensional space. When multiple lines intersect at a common point, that point of intersection represents the solution to the system.
Why would we use a system of linear equations? They are helpful in numerous fields such as economics, engineering, and physics, as they allow us to find solutions that satisfy multiple conditions simultaneously.
To solve these systems, methods such as graphing, substitution, elimination, or matrix techniques are used. For larger systems, matrix representation is preferred as it can simplify handling complex operations.

Matrix Representation

Matrix representation is a convenient and compact way of handling systems of linear equations. By converting equations into a matrix form, we can use matrix operations to solve them more efficiently.
Consider our given system of equations:

• The variables and their associated coefficients form the bulk of the matrix.
• An augmented matrix is created by including the constants from each equation as an additional column.

The matrix version of the system looks like this:\[\begin{bmatrix}3 & -2 & | & 4 \2 & 0 & | & 3 \-1 & 9 & | & 8 \5 & -7 & | & 13\end{bmatrix}\]This matrix representation makes it easy to perform row operations, which are essential for solving the system using matrix techniques like Gaussian elimination.

Coefficient Matrix

The coefficient matrix consists of all the coefficients of the variables from our system of linear equations. It's the part of the augmented matrix that excludes the constants on the right side of the equations.
For our system, the coefficient matrix is:\[\begin{bmatrix}3 & -2 \2 & 0 \-1 & 9 \5 & -7\end{bmatrix}\]Why is the coefficient matrix important?

• It is essential for determining the uniqueness or existence of solutions in a system.
• It can be used to construct the identity matrix for solving using inverse methods when applicable.

Understanding the coefficient matrix simplifies the complexity of systems and aids in efficient calculation of solutions.

Linear Algebra Concepts

Linear algebra provides the tools necessary to manage and solve systems of linear equations effectively. It involves various concepts and operations foundational to understanding matrices.
Some key linear algebra concepts relevant to this exercise include:

• Matrix addition and subtraction: Useful when combining systems.
• Scalar multiplication: Multiplies every element within a matrix by a number.
• Row Echelon Form (REF) and Reduced Row Echelon Form (RREF): Forms of a matrix that help in determining solutions to a system.

Matrix operations enable efficient computational techniques like Gaussian or Gauss-Jordan elimination, which are fundamental algorithms to solve systems of linear equations.
By appreciating these concepts, you gain powerful methods to tackle more complex systems encountered in higher studies and various applications.