Ok.. Have a look at this problem and then look at the second post:
Can anyone explain the second examiners solution? I understand the first and third approach but what is happening in the second method below, In particular what does trianglex, triangle y , trianglez mean in the second line of the solution i have circled in red?
The "$\displaystyle \Delta$" is, of course the determinant. Since that is 0 it follows that there cannot be a unique solution. But I think when they say "Showig $\displaystyle \Delta= 0$ and thinking this is it" they the mean that alone does NOT show that the system is inconsistent- there might be an infinite number of solutions. I do think that "and $\displaystyle \Delta_x$, $\displaystyle \Delta_y$, $\displaystyle \Delta_z$= 0" is misleading. In order to have the system inconsistent, you must show that $\displaystyle \Delta= 0$ and that at least one of $\displaystyle \Delta_x$, $\displaystyle \Delta_y$, $\displaystyle \Delta_z$ is NOT 0.
I am rather disappointed that the most direct demonstration is not given:
If you add the second and third equations, the "y" and "-y" terms cancel leaving 11x+ 11z= 37.
Multiplying the second equation by 2 gives 6x+ 2y- 8z= 14 and adding the first equation to that gives 7x+ 7z= 14. Do you see why those two equations are impossible?