Problem 3 - Advanced Engineering Mathematics

Question

Use either Gaussian elimination or Gauss-Jordan elimination to solve the given system or show that no solution exists. \(9 x_{1}+3 x_{2}=-5\) \(2 x_{1}+x_{2}=-1\)

Step-by-Step Solution

Answer

The system of equations has no solution.

Step 1: Write the System of Equations in Augmented Matrix Form

The given system of equations is \( 9x_1 + 3x_2 = -5 \) and \( 2x_1 + x_2 = -1 \). First, write this system as an augmented matrix:\[\begin{bmatrix}9 & 3 & | & -5 \2 & 1 & | & -1\end{bmatrix}\]

Step 2: Perform Row Operations to Achieve Row Echelon Form

To eliminate the first term of the second equation, multiply the first row by \(2\) and the second row by \(9\) and subtract the second row from the first row.\( 2R_1 - 9R_2 \rightarrow R_1 \)This results in:\[\begin{bmatrix}0 & 0 & | & 1 \2 & 1 & | & -1 \end{bmatrix}\]

Step 3: Interpret the Result

The row operation has led to a row of the form \( \begin{bmatrix} 0 & 0 & | & 1 \end{bmatrix} \) in the matrix. This indicates an inconsistency, meaning that the system of equations does not have a solution.

Key Concepts

Gauss-Jordan elimination

Gauss-Jordan elimination is a method used to solve systems of linear equations. It involves performing row operations on an augmented matrix to transform it into reduced row echelon form (RREF). When this form is achieved, the solutions to the system are readily apparent.

Row operations include:

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

Gauss-Jordan elimination goes beyond simply reaching row echelon form by continuing to eliminate variables in all rows, not just below each pivot. The final result is a diagonal of leading 1s, which clearly shows the solutions to the system if they exist. If the process results in a row with all zero coefficients, aside from a non-zero value on the right side of the augmentation bar, it indicates an inconsistency, meaning the system has no solution.

Augmented matrix

An augmented matrix is a streamlined way to represent a system of linear equations. It combines the coefficients of the variables with the constants from the right side of the equations into a single matrix. This makes it easier to apply row operations and manipulate the equations systematically.

For a system:

• 9x₁ + 3x₂ = -5
• 2x₁ + x₂ = -1

The augmented matrix is written as:\[\begin{bmatrix} 9 & 3 & | & -5 \ 2 & 1 & | & -1 \end{bmatrix}\]The bar on the right separates the coefficients from the constants. Transforming this matrix using row operations is what helps to solve the system using Gaussian or Gauss-Jordan elimination techniques.

Row echelon form

Row echelon form (REF) is a key stage in solving systems of equations using matrix methods. A matrix is in row echelon form when:

• All non-zero rows are above any rows of all zeros.
• The leading coefficient (first non-zero number from the left, also called a pivot) of a non-zero row is always strictly to the right of the leading coefficient of the row above it.
• All entries in a column below a pivot are zeros.

Achieving row echelon form is a critical step in both Gaussian and Gauss-Jordan elimination. For Gaussian elimination alone, once REF is achieved, back-substitution can be used if the matrix represents a system with one unique solution. However, sometimes reaching this form is enough to reveal inconsistencies, as seen in matrices representing systems without solutions.

System of equations

A system of equations refers to a set of equations with multiple variables that are solved simultaneously. The aim is to find values for the variables that satisfy all the equations at once. There are three possible outcomes:

• A unique solution, where one specific set of values satisfies all equations.
• Infinitely many solutions, where several sets of values can satisfy the equations.
• No solution, indicating the equations are inconsistent and cannot be satisfied by any set of values.

In the given exercise, the system is linear, and by applying Gaussian elimination, we discovered it has no solution. The row operations led us to a row in the echelon form that indicated a contradiction (like 0 = 1), revealing the inconsistency of the system. Understanding when and why a system doesn't have a solution is crucial in fields that rely on mathematical modeling and linear algebra.