Answer to Question #178268 in Linear Algebra for sarang

Question

Find the conditions on a, b, and c so that the following linear system of equations have a solution.

−2x + y + z = a

x − 2y + z = b

x + y − 3z = c

Answer 1

$\Delta=-2\cdot(6-1)-1\cdot(-3-1)+1\cdot(1+2)=-10+4+3=-3≠0,$

$\Delta_1=a\cdot(6-5)-1\cdot(-3b-c)+1\cdot(b+2c)=a+4b+3c,$

$\Delta_2=-2\cdot(-3b-c)-a\cdot(-3-1)+1\cdot(c-b)=4a+5b+3c,$

$\Delta_3=-2\cdot(-2c-b)-1\cdot(c-b)+a\cdot(1+2)=3a+3b+3c,$

$x=\frac{\Delta_1}{\Delta}=-\frac a3-\frac 43 b-c,$

$y=\frac{\Delta_2}{\Delta}=-\frac 43 a-\frac 53 b-c,$

$z=\frac{\Delta_3}{\Delta}=-a-b-c,$

there is a solution for any a, b, c.