Inconsistent System
A system is this if it has no solution
An inconsistent system is a set of linear equations that have no solution. In other words, the equations in the system are contradictory and cannot be satisfied simultaneously. Graphically, an inconsistent system represents a set of lines that do not intersect at any point. Mathematically, we can determine if a system is inconsistent by looking at the row echelon form or reduced row echelon form of its augmented matrix. If the augmented matrix reduces to a row of zeros where the last entry is nonzero, then the system is inconsistent. Alternatively, if a pivot (leading entry) is missing in one of the columns of the augmented matrix, then the system is inconsistent.
For example, consider the system of equations:
3x + 2y = 7
6x + 4y = 9
We can write the corresponding augmented matrix as:
[3 2 | 7]
[6 4 | 9]
By performing row operations, we can obtain the row echelon form of the augmented matrix as:
[3 2 | 7]
[0 -2 | -5]
Notice that the second row implies that -2y = -5, which means that y = 5/2. However, the first row implies that 3x + 2(5/2) = 7, which simplifies to 3x + 5 = 7 or 3x = 2. So, x = 2/3. However, if we substitute these values into the second equation, we get:
6(2/3) + 4(5/2) = 4/3 + 10 = 34/3
This contradicts the right-hand side of the equation, which is 9. Therefore, this system has no solution and is inconsistent.
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