Question

Create a system of linear equations that when solved algebraically, prove to have no solution.


WRITER

Answer 1

One simple inconsistent system is:

x + y = 1
2x + 2y = 3

Algebraic proof by elimination:
- Multiply the first equation by 2: 2x + 2y = 2.
- Subtract this from the second equation: (2x + 2y) - (2x + 2y) = 3 - 2, which gives 0 = 1.

Since 0 = 1 is a contradiction, the system has no solution.

(Equivalently, in augmented-matrix form:
[1 1 | 1]
[2 2 | 3]
→ R2 ← R2 − 2R1 gives
[1 1 | 1]
[0 0 | 1],
which is inconsistent.)