Solving System for Non-trivial Solutions

We need to find the non-trivial solutions for the system of linear equations:

• Equation 1: $x+y-6z=0$
• Equation 2: $-3x+y+2z=0$
• Equation 3: $x-y+2z=0$

Non-trivial solutions are solutions where $x, y,$ and $z$ are not all zero. We can verify the options by substituting the proposed solution forms into the equations.

Verifying Non-trivial Solution Set

Let's test the form $x=2c, y=4c, z=c$, where $c \neq 0$. This represents a potential non-trivial solution.

1. Check Equation 1: Substitute $x=2c, y=4c, z=c$ into $x+y-6z=0$: $(2c) + (4c) - 6(c) = 6c - 6c = 0$ This equation is satisfied.
2. Check Equation 2: Substitute $x=2c, y=4c, z=c$ into $-3x+y+2z=0$: $-3(2c) + (4c) + 2(c) = -6c + 4c + 2c = 0$ This equation is satisfied.
3. Check Equation 3: Substitute $x=2c, y=4c, z=c$ into $x-y+2z=0$: $(2c) - (4c) + 2(c) = 2c - 4c + 2c = 0$ This equation is satisfied.

Since the expressions $x=2c, y=4c, z=c$ satisfy all three equations for any non-zero scalar $c$, this represents the set of non-trivial solutions to the system.