Problem 9 - Advanced Engineering Mathematics

Question

In Problems 1-20, use either Gaussian elimination or Gauss-Jordan elimination to solve the given system or show that no solution exists. $$ \begin{array}{r} x_{1}-x_{2}-x_{3}=8 \\ x_{1}-x_{2}+x_{3}=3 \\ -x_{1}+x_{2}+x_{3}=4 \end{array} $$

Step-by-Step Solution

Step 1: Write the augmented matrix

First, we need to convert the given system of equations into an augmented matrix. For the system:\[\begin{align*}x_1 - x_2 - x_3 &= 8,\x_1 - x_2 + x_3 &= 3,\-x_1 + x_2 + x_3 &= 4,\end{align*}\]the augmented matrix is:\[\begin{bmatrix}1 & -1 & -1 & | & 8\1 & -1 & 1 & | & 3\-1 & 1 & 1 & | & 4\end{bmatrix}\].

Key Concepts

Gaussian Elimination

When dealing with systems of equations, Gaussian Elimination is a fundamental technique used to find the solutions, if they exist. This method involves systematically reducing a matrix to an upper triangular form using row operations. These operations include:

• Swapping two rows
• Multiplying a row by a non-zero scalar
• Adding or subtracting a multiple of one row to another row

The goal is to have all zeros below the leading coefficients (or pivot elements). Once in upper triangular form, you can use back substitution to find the values of the variables.
By progressing in a step-by-step approach, each row reveals valuable information about the variable it represents, leading to the eventual resolution of the system.

Gauss-Jordan Elimination

Gauss-Jordan Elimination takes the process of Gaussian Elimination further by transforming the matrix into reduced row echelon form (RREF). In this form, each leading entry (or pivot) in a row is a 1, and is the only non-zero entry in its column. The steps include:

• Converting the leading coefficients to 1 by dividing the row by the pivot element
• Eliminating all other entries in the pivot column by scaling and adding the pivot row to others

The advantage of using Gauss-Jordan Elimination over Gaussian Elimination is that once the matrix is in RREF, the solutions are immediately apparent without needing back substitution. Therefore, it is often preferred when solving problems where an explicit solution is required directly from the matrix notation.

Systems of Equations

A system of equations consists of multiple equations that are solved together because they share common variables. To find a solution, we look for values of the variables that satisfy all equations simultaneously.
Solving systems of equations is a central problem in linear algebra, as it applies to numerous practical situations, from engineering to economics. There are three possible outcomes when solving a system:

• A unique solution exists, where all lines or planes intersect at a single point
• Infinitely many solutions exist, often when the same line or plane is represented multiple times
• No solution exists, implying parallel lines or planes that never meet

Understanding these outcomes is crucial for properly interpreting the results of the elimination processes.

Augmented Matrix

An augmented matrix is an essential tool in solving systems of equations, serving as a condensed form of the system. It involves writing the coefficients of the variables and the constant terms from the system's equations in matrix form, separating with a bar ( | ) to indicate the constant terms column.
For example, given the system:\[\begin{align*}x_1 - x_2 - x_3 &= 8,\x_1 - x_2 + x_3 &= 3,\-x_1 + x_2 + x_3 &= 4,\end{align*}\]The augmented matrix is:\[\begin{bmatrix}1 & -1 & -1 & | & 8\1 & -1 & 1 & | & 3\-1 & 1 & 1 & | & 4\end{bmatrix}\]Using an augmented matrix, eliminating rows to reach reduced row echelon form simplifies solving systems significantly. It is a visual and structural representation that makes complex systems more manageable and is the starting point for both Gaussian and Gauss-Jordan elimination methods.