Solving Systems of Equations: Gaussian and Gauss-Jordan Elimination

Systems of Equations and Inequalities

Solving Systems Using Gaussian and Gauss-Jordan Elimination

Systems of linear equations can be solved using various algebraic methods. Two important techniques are Gaussian elimination and Gauss-Jordan elimination, which use matrices to systematically reduce the system to a form where the solutions can be easily identified.

• Gaussian Elimination: This method transforms the system's augmented matrix into an upper triangular form using row operations, allowing for back-substitution to find the solutions.

• Gauss-Jordan Elimination: This method continues the process to obtain a reduced row-echelon form, where the solution can be read directly from the matrix without back-substitution.

Key Terms and Definitions

• System of Equations: A set of two or more equations with the same variables.

• Augmented Matrix: A matrix that includes the coefficients and constants from a system of equations.

• Row Operations: Operations that can be performed on the rows of a matrix: swapping rows, multiplying a row by a nonzero constant, and adding or subtracting multiples of one row to another.

Example Problem

Solve the following system using Gaussian or Gauss-Jordan elimination:

• $-4x + 2y = 20$

• $x + y = -5$

Step 1: Write the Augmented Matrix

$\begin{bmatrix} -4 & 2 & | & 20 \\ 1 & 1 & | & -5 \end{bmatrix}$

Step 2: Use Row Operations to Achieve Upper Triangular or Reduced Row-Echelon Form

1. Multiply the second row by 4 and add to the first row to eliminate $x$ from the first equation:

Row 1: $-4x + 2y = 20$ Row 2: $x + y = -5$

Multiply Row 2 by 4: $4x + 4y = -20$ Add to Row 1: $(-4x + 2y) + (4x + 4y) = 20 + (-20)$ $0x + 6y = 0$ So, the new system is:

$\begin{bmatrix} 0 & 6 & | & 0 \\ 1 & 1 & | & -5 \end{bmatrix}$

2. Solve for $y$ from the first row: $6y = 0 \implies y = 0$

3. Substitute $y = 0$ into the second equation: $x + 0 = -5 \implies x = -5$

Solution Set:

$\boxed{\begin{bmatrix} -5 \\ 0 \end{bmatrix}}$

Summary Table: Comparison of Gaussian and Gauss-Jordan Elimination

Method	Form Achieved	Solution Process
Gaussian Elimination	Upper Triangular	Back-substitution required
Gauss-Jordan Elimination	Reduced Row-Echelon	Solution read directly from matrix

Additional info: Both methods are foundational for solving larger systems and are widely used in algebra and linear algebra courses.