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First, nope, all homogeneous systems are not inconsistent. To show why, I would need to get into matrices and their relation to systems.
Take for instance:
2x+5y = 0
3x - 7y = 0
Only as an example, to show the relation to matrices. This can be written in matrix form as:
| 2 5 | * | x | = | 0 |
| | | | | |
| 3 -7 | | y | | 0 |
Sorry, "|" is the best matrix symbols I can find. Shortening it, we would write it as:
Ax = 0
Where A is the coefficient matrix, "x" is the matrix with x and y, and "0" is the matrix with the zeros.
Now, we will get general, using only Ax=0. One may consider that, given this equation, x would have to be equal to 0. But, recall, the determinant of A could be 0, also. If det |A| = 0, then x could be anything, making the system inconsistent.
I hope this helps. Good luck.
A homogeneuos system of equation is always consistent because it has always at least one solution i.e. trivial solution. X=Y=Z=......=0
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