Systems of Equations and Gaussian Elimination | Generated by AI

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This tutorial provides a detailed guide on systems of equations, their representation, solutions, and the Gaussian elimination method used to solve them.


1. Systems of Equations: Definition and Representation

A system of linear equations consists of multiple linear equations that share variables. A general system with \( n \) variables and \( m \) equations can be written as:

[ \begin{aligned} a_{11}x_1 + a_{12}x_2 + \dots + a_{1n}x_n &= b_1
a_{21}x_1 + a_{22}x_2 + \dots + a_{2n}x_n &= b_2
\vdots &
a_{m1}x_1 + a_{m2}x_2 + \dots + a_{mn}x_n &= b_m \end{aligned} ]

where:

Matrix Representation

A system of equations can be represented using matrices:

[ A \mathbf{x} = \mathbf{b} ]

where:

The augmented matrix is written as:

[ [A | \mathbf{b}] ]

Example:
[ \begin{aligned} 2x + 3y &= 8
5x - y &= 3 \end{aligned} ]

Matrix representation:
[ \begin{bmatrix} 2 & 3
5 & -1 \end{bmatrix} \begin{bmatrix} x
y \end{bmatrix} = \begin{bmatrix} 8
3 \end{bmatrix} ]

Augmented matrix:
[ \left[ \begin{array}{cc|c} 2 & 3 & 8
5 & -1 & 3 \end{array} \right] ]


2. Gaussian Elimination Method

Gaussian elimination is a systematic method for solving systems of equations by transforming the augmented matrix into row echelon form (REF) and then solving for the variables using back-substitution.

Steps of Gaussian Elimination

  1. Convert the augmented matrix into an upper triangular (row echelon) form by using row operations:
    • Swap rows if needed.
    • Multiply a row by a nonzero constant.
    • Add or subtract a multiple of one row from another.
  2. Back-substitution to find the solution.

Example 1: Solving a System using Gaussian Elimination

Solve the system:
[ \begin{aligned} 2x + y - z &= 3
4x - 6y &= 2
-2x + 7y + 2z &= 5 \end{aligned} ]

Step 1: Convert to Augmented Matrix

[ \left[ \begin{array}{ccc|c} 2 & 1 & -1 & 3
4 & -6 & 0 & 2
-2 & 7 & 2 & 5 \end{array} \right] ]

Step 2: Make the First Pivot 1

Divide row 1 by 2: [ \left[ \begin{array}{ccc|c} 1 & 0.5 & -0.5 & 1.5
4 & -6 & 0 & 2
-2 & 7 & 2 & 5 \end{array} \right] ]

Step 3: Eliminate First Column Below Pivot

Replace row 2 by subtracting 4 times row 1:
Replace row 3 by adding 2 times row 1:

[ \left[ \begin{array}{ccc|c} 1 & 0.5 & -0.5 & 1.5
0 & -8 & 2 & -4
0 & 8 & 1 & 8 \end{array} \right] ]

Step 4: Make Second Pivot 1

Divide row 2 by -8:

[ \left[ \begin{array}{ccc|c} 1 & 0.5 & -0.5 & 1.5
0 & 1 & -0.25 & 0.5
0 & 8 & 1 & 8 \end{array} \right] ]

Step 5: Eliminate Second Column Below Pivot

Replace row 3 by subtracting 8 times row 2:

[ \left[ \begin{array}{ccc|c} 1 & 0.5 & -0.5 & 1.5
0 & 1 & -0.25 & 0.5
0 & 0 & 3 & 4 \end{array} \right] ]

Step 6: Back Substitution

Solve from bottom to top:

Solution: [ x = \frac{11}{6}, \quad y = \frac{7}{6}, \quad z = \frac{4}{3} ]


3. Solution Types

  1. Unique Solution: A single solution for \( x_1, x_2, \dots \).
  2. Infinite Solutions: Many solutions (system is dependent).
  3. No Solution: Inconsistent system.

4. Homogeneous vs. Non-Homogeneous Systems


Conclusion

Let me know if you need more examples or explanations! 🚀


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