Systems of Linear Equations

Systems of linear equations arise in many fields of science and engineering.

eg. Robotics - Force equations describing the motion of each link in a robot arm form a systems of linear equations.

A simple 2x2 system of linear equations has 2 unknowns or variables, and 2 equations like

2 x + 3 y = 6
4 x  - 6 y = 2


In electronics, systems of linear equations are important in the analysis of electronics networks:

Application of "loop analysis" to the above network will result in the following set of linear equations:

3x3 system

3  I₁  - 1 I₂ + 0 I₃ = 10
-1 I₁ + 5 I₂  - 2 I₁ = 0
0  I₁  - 2 I₂ + 8 I₃ = -5

where the unknowns or variables are the "mesh currents" I₁ = 3.5A,   I2 = 0.5A, and I3 = -0.5A.


In a linear system, the power of all variables is 1 and all the coefficients are constants.

 eg.
a₁  x + b₁ y = c₁ 
a₂  x + b₂ y = c₂

a₁, b₁, c₁, a₂, b₂, c₂ - all constants
x, y - variables raised to power 1

A solution of the above system is any pair (x0, y0) which simultaneously satisfies both equations.

I general, a linear system of N equations in N unknowns will have:
1. Exactly one solution
2. No solution (Inconsistent)
3. Infinitely many solutions (Dependent or Redundant system)

For example: 2x2 systems...

1.

==> one solution


2.
 ==> No solution

3.

  ==>  Infinitely many solutions


Various Methods of Solving Linear System of Equations:
1. Graphing

2. Substitution

3. Elimination
4. Gauss and Gauss-Jordan Elimination
5. Determinants (Cramer's Rule)
6. [Matrices]