I just can't figure this out. aaaaah!!!

Get y by itself in the first equation

 

x + y + 3z = 10      →   y =  10 - x - 3z

 

Substitute this into the second equation and simplify:

 

-4x + 2y +5z = 7    →   -4x + 2(10 - x - 3z) + 5z = 7   →   -4x + 20 -2x - 6z + 5z  = 7  →  -6x - z = -13  →

 

6x + z  = 13

 

So we have this system:

 

k*x + z = 3

 

6x + z = 13 

 

And notice that, if k =6, we would have this system:

 

6x + z  = 3      and

 

6x + z  = 13       but this is impossible, because two things added together can't possibly have two different outcomes 

 

So....as geno found, when k = 6, the system has no solution...!!!

 

WORD OF CPhill