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. infinite number of solutions if $a+b+c=0$
• D. none of these

Answer 1

Answer: (A), (C)

The correct options are
A no solution if $a+b+c\ne 0$
C infinite number of solutions if $a+b+c=0$

Let

$-2x+y+z=a$
$x-2y+z=b$
$x+y-2z=c$

When we add three equations, we get

$0=a+b+c$.

Thus, the system does not have a solution if $a+b+c\ne 0$.

If $a+b+c=0$, the system has infinite number of solutions given by

$x=\frac{1}{3}(3k-2a-b)$

$y=\frac{1}{3}(3k-a-2b)$

$z=k$

where $k\in C$.

Hence, options A and C.