An inconsistent system is a system of linear equations with two or more variables, the solution of which leads to at least one variable having ambiguous values (more than one value) or an inequality. This means that a unique solution is not possible. For example, x+y+z=10 and 2x+2y+2z=11. This would lead to 20=11, which can never be true.
Another example is x+y=3, x+z=4, 2x+y+z=8. If we add the first two equations we get 2x+y+z=7, but the third equation gives a sum of 8, and 7 does not equal 8.