Question

If the equation $-2x+y+z=l;-2y+z=m;x+y-2z=n$ such that $l+m+n=0,$ then the system has

• A. a non-zero unique solution
• B. trivial solution
• C. infinitely many solution
• D. no solution

Answer 1

The correct option is C infinitely many solution

Add given three equations, we get $-x=l+m+n=0$ $($ Given $l+m+n=0)$

Therefore $x=0$ $($which implies $yz$-plane$)$

Put $x=0$ in given equations , we get $y+z=l$ , $-2y+z=m$ and $y-2z=n$

Subtracting the second equation from first, we get $y=\frac{l-m}{3}$ and subtracting the third equation from first, we get $z=\frac{l-n}{3}$

Since $l,m,n$ can be anything which satisfies $l+m+n=0$ , $y,z$ can have infinite values

So the given system has infinite number of solution.