Systems of Equations and Gaussian Elimination | Generated by AI
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:
- \( x_1, x_2, \dots, x_n \) are the unknown variables.
- \( a_{ij} \) are the coefficients.
- \( b_1, b_2, \dots, b_m \) are the constants on the right-hand side.
Matrix Representation
A system of equations can be represented using matrices:
[ A \mathbf{x} = \mathbf{b} ]
where:
-
\( A \) is the coefficient matrix:
[ A = \begin{bmatrix} a_{11} & a_{12} & \dots & a_{1n}
a_{21} & a_{22} & \dots & a_{2n}
\vdots & \vdots & \ddots & \vdots
a_{m1} & a_{m2} & \dots & a_{mn} \end{bmatrix} ] -
\( \mathbf{x} \) is the variable column vector:
[ \mathbf{x} = \begin{bmatrix} x_1
x_2
\vdots
x_n \end{bmatrix} ] -
\( \mathbf{b} \) is the constant column vector:
[ \mathbf{b} = \begin{bmatrix} b_1
b_2
\vdots
b_m \end{bmatrix} ]
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
- 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.
- 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:
- \( 3z = 4 \Rightarrow z = \frac{4}{3} \)
- \( y - 0.25z = 0.5 \Rightarrow y = 0.5 + 0.25(4/3) = \frac{7}{6} \)
- \( x + 0.5y - 0.5z = 1.5 \Rightarrow x = 1.5 - 0.5(7/6) + 0.5(4/3) = \frac{11}{6} \)
Solution: [ x = \frac{11}{6}, \quad y = \frac{7}{6}, \quad z = \frac{4}{3} ]
3. Solution Types
- Unique Solution: A single solution for \( x_1, x_2, \dots \).
- Infinite Solutions: Many solutions (system is dependent).
- No Solution: Inconsistent system.
4. Homogeneous vs. Non-Homogeneous Systems
- Homogeneous system: \( A\mathbf{x} = 0 \)
- Always has at least the trivial solution \( \mathbf{x} = 0 \).
- May have infinitely many solutions if the determinant of \( A \) is zero.
- Non-homogeneous system: \( A\mathbf{x} = \mathbf{b} \)
- May have unique, infinite, or no solutions.
Conclusion
- Gaussian elimination is a powerful method to solve systems.
- The number of solutions depends on the rank of the matrix.
- Homogeneous systems always have a trivial solution.
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