Problem 19 - 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. $$ \begin{aligned} x_{2}+x_{3}-x_{4} &=4 \\ x_{1}+3 x_{2}+5 x_{3}-x_{4} &=1 \\ x_{1}+2 x_{2}+5 x_{3}-4 x_{4} &=-2 \\ x_{1}+4 x_{2}+6 x_{3}-2 x_{4} &=6 \end{aligned} $$

Step-by-Step Solution

Answer

No solution exists for the given system.

Step 1: Write the Augmented Matrix

Convert the given system of equations into an augmented matrix. Each row corresponds to an equation, and each column corresponds to the coefficients of the variables and the constant terms: \[\begin{bmatrix}0 & 1 & 1 & -1 & | & 4 \1 & 3 & 5 & -1 & | & 1 \1 & 2 & 5 & -4 & | & -2 \1 & 4 & 6 & -2 & | & 6\end{bmatrix}.\]

Step 2: Apply Row Reduction - Making Row 1 a Leading 1

Divide the first row by the coefficient 1 to make the leading coefficient of the second column a 1. Note that the first column starts with 0, hence move to the second column for pivoting.\[\begin{bmatrix}0 & 1 & 1 & -1 & | & 4 \1 & 3 & 5 & -1 & | & 1 \1 & 2 & 5 & -4 & | & -2 \1 & 4 & 6 & -2 & | & 6\end{bmatrix}\]

Step 3: Eliminate x₂ from Rows 2, 3, and 4

Subtract the first row (row oriented on x₂) multiplied by appropriate constants from the other rows to eliminate the x₂ term in rows 2, 3, and 4. After such operations, the matrix looks like:\[\begin{bmatrix}0 & 1 & 1 & -1 & | & 4 \1 & 0 & 2 & 2 & | & -11 \1 & 0 & 3 & -3 & | & -10 \1 & 0 & 2 & 2 & | & -10\end{bmatrix}\]

Step 4: Apply Row Reduction - Adjust Row 2

Now make the first row under x₁ into a leading one. Subtract the third row by the second to further simplify the system.\[\begin{bmatrix}0 & 1 & 1 & -1 & | & 4 \1 & 0 & 2 & 2 & | & -11 \1 & 0 & 3 & -3 & | & -10 \0 & 0 & -1 & 0 & | & 1\end{bmatrix}\]

Step 5: Back-Substitute for Variables

From the final row, which is now 0 = 1 (inconsistent), recognize that no values of variables will satisfy this equation. Therefore, the entire system has no solution.

Key Concepts

Gauss-Jordan elimination

When tackling systems of linear equations, the Gauss-Jordan elimination method is a powerful tool. It extends Gaussian elimination by transforming the augmented matrix into reduced row-echelon form. This means each pivot is a 1 and all other entries in the pivot's column are 0.
The process involves a series of row operations, including swapping rows, multiplying a row by a non-zero scalar, and adding or subtracting rows.

• These operations aim to clear columns by creating zeros below each leading coefficient.
• Ultimately, the matrix is simplified enough to straightforwardly read the solution.

This method is robust for finding unique solutions when they exist, or for identifying when no solution is possible, as in the case of an inconsistent system.

augmented matrix

An augmented matrix is a convenient tool in linear algebra used to represent a system of linear equations. It combines the coefficient matrix with the constant terms from the equations' right-hand sides into a single matrix.
This inclusion of the constants facilitates solving the system using elimination methods.

• Each row in the augmented matrix corresponds to an equation from the system.
• The columns represent the coefficients of the variables, while the last column contains the constants.

Writing an equation system as an augmented matrix allows us to use systematic row operations to find solutions or determine the nature of the solution set.

row reduction

Row reduction refers to the process of performing row operations to simplify an augmented matrix to either echelon form or reduced row-echelon form. This process involves strategically manipulating the matrix to simplify the system for solving.
Key operations include:

• Row swapping for repositioning.
• Scaling rows to achieve pivot positons.
• Adding or subtracting rows to clear columns of unwanted coefficients.

The goal of row reduction is to achieve a row-echelon form where it's easier to back-substitute and solve for the variables.
Throughout, keeping track of operations ensures accuracy in deriving insights from the given system.

inconsistent system of equations

An inconsistent system of equations is one where no set of variable values will simultaneously satisfy all equations within the system.
This situation typically arises when, during the row reduction process, an augmented matrix reveals a row where the coefficients are all zeros but the constant term is non-zero, like "0 = 1".

• This indicates a contradiction inherent in the system.
• It suggests that the original equations are at odds, for example, having parallel or coinciding lines that never intersect.

Recognizing an inconsistent system helps prevent futile attempts at finding non-existent solutions and highlights the need for revisiting the system's setup or assumptions.