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 = 16x + 20y + 22z = 26x + 4y + 8z = 3using Gaussian elimination.

Solution First, we convert the system to an

augmented matrix. The result is

Solution

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).