RECOGNIZING INCONSISTENT LINEAR SYSTEMS

What is an Inconsistent Linear System?  An inconsistent linear system is a system of equations that has no solutions  Essentially, the equations contradict each other.  Attempting to solve an inconsistent linear system leads to a result like 1 = 2: an impossibility.

Inconsistent Linear Systems and Gaussian Elimination  How can we recognize an inconsistent system when we are using Gaussian elimination?  If, as you eliminate variables, you ever end up with a row where all of the coefficients are zero and the constant term is not, you have an inconsistent linear system.  If the system is inconsistent, these rows will appear naturally through the Gaussian elimination process.

Example  Solve the system of equations 2x + 4y + 5z = 1 6x + 20y + 22z = 2 6x + 4y + 8z = 3 using Gaussian elimination.

Solution  First, we convert the system to an augmented matrix. The result is

Solution

Note  In general, if two of your rows of coefficients are ever constant multiples of one another, you either have infinitely many solutions (if adding appropriate combinations of two rows results in the equation 0 = 0) or no solutions (if adding appropriate combinations of the two rows results in a contradiction).