SOLUTION: Show that the following system is consistent if and only if c=2a-3b and solve the system in this case 2x-y+3z=a 3x+y-5z=b -5x-5y+21z=c

We procede as though we were going to solve the
system by the Gauss-Jordan method:

The augmented matrix is



Get a 0 where the 3 is on the 2nd row 1st column,
by row operation -3R1+2R2->R2



Get a 0 where the -5 is on the 3rd row 1st column,
by row operation 5R1+2R3->R3



Get a 0 where the -15 is on the 3rd row 2nd column,
by row operation 3R2+R3->R3



That last row is equivalent to the equation



That can be true and thus the system is 

consistent if and only if



Solving for c gives





I'll let you finish solving the system.
If you have trouble, tell me in the 
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charge because I just do this as a hobby.

Edwin