Consistency of Linear Systems A linear system of equations is a set of linear equations with the same number of equations as variables. Solving a linear syst...
A linear system of equations is a set of linear equations with the same number of equations as variables. Solving a linear system means finding the values of the variables that make all the equations true.
There are three main types of linear systems:
Consistent: This means that the system has exactly one solution.
Inconsistent: This means that the system has no solution.
Singular: This means that the system has exactly two solutions (which are linearly dependent).
The consistency of a linear system is determined by the determinant of the coefficient matrix:
If the determinant is zero, the system is consistent.
If the determinant is non-zero, the system is inconsistent.
If the determinant is positive, the system is consistent and has one unique solution.
If the determinant is negative, the system is inconsistent and has no unique solution.
Here's an example:
Consider the following system of linear equations:
x + y = 3
x - y = 1
The determinant of this system is:
(1)(1) - (-1)(-1) = 1 - 1 = 0
Since the determinant is zero, the system is consistent and has exactly one solution. This solution is unique.
Additional notes:
A system with more variables than equations will always be consistent.
A system with more equations than variables will always be consistent.
A system with one equation and n variables will always be consistent (where n is the number of variables)