Linear system what does H not equal

Question

Let k, h be unknown constants and consider the linear system:

x - 4y + 7z = -7

7x - 4y +8z = 3

17x - 20y + hz = k



This system has a unique solution whenever h does not equal ???

Answer 1

the system does not have a unique or any solution

when h=40 and k=15

or

when h=35 and k=-35

because then the last equation and one of the other equations are contradictory, parallel graphically with no intersection,

h=40 and k=15

makes the last equation

17x -20y +40z = 15 which contradicts the 2nd equation: 7x-4y+8z=3 equivalent to 35x-20y+40x=15 when multiplied by 5; unless x=0 but then there are an infinite number of solutions not any unique solution

h=35 and k=-35

makes the last equation

17x-20y +35z = -35 which contradicts the 1st equation: 4x-4y +7z =-7 equivalent to 20x-20y+35z =-35 when multiplied by 5; unless x=0 but then there are an infinite number of solutions, not any unique solution