Question

The system of equations
$-2x+y+z=a,x-2y+z=b,x+y-2z=c$ has:

• A. no solution if $a+b+c\ne 0$
• B. unique solution if $a+b+c=0$
• C. infinte number of solutions if $a+b+c=0$
• D. infinte solution if $a+b+c\ne 0$

Answer 1

Answer: (A), (C)

infinte number of solutions if $a+b+c=0$

$\Delta =\left|\begin{array}{ccc}-2& 1& 1\\ 1& -2& 1\\ 1& 1& -2\end{array}\right|=0$

${\Delta }_x=\left|\begin{array}{ccc}a& 1& 1\\ b& -2& 1\\ c& 1& -2\end{array}\right|=3a+3b+3c$

${\Delta }_y=\left|\begin{array}{ccc}-2& a& 1\\ 1& -b& 1\\ 1& c& -2\end{array}\right|=3a+3b+3c$

${\Delta }_z=\left|\begin{array}{ccc}-2& 1& a\\ 1& -2& b\\ 1& 1& c\end{array}\right|=3a+3b+3c$