What Should Be Written for x3 in a Homogeneous System When x1 and x2 are Zero?

[MoreDrinks]

Homework Statement

Solve the homogeneous system of equations.

Homework Equations

The relevant matrix is like so:
1 0 0
-1 0 0
3-5 0

The Attempt at a Solution

Add R1 to R2, then add -3R1 to R3.

1 0 0
0 0 0
0-5 0

Interchange R2 and R3, then divide the new R2 by -1/5

1 0 0
0 1 0
0 0 0

Under other circumstances where there's a general solution to such a matrix, with a row of zeroes on the bottom, but not an empty column for x₃, you would write x₃=r or what have you and then include r when solving for x₁ and x₂. In this case, where x₁ and x₂ simply equal zero, what would one write about x₃?

 

[Mark44]

Homework Statement

Solve the homogeneous system of equations.

Homework Equations

The relevant matrix is like so:
1 0 0
-1 0 0
3-5 0

The Attempt at a Solution

Add R1 to R2, then add -3R1 to R3.

1 0 0
0 0 0
0-5 0

Interchange R2 and R3, then divide the new R2 by -1/5

1 0 0
0 1 0
0 0 0

Under other circumstances where there's a general solution to such a matrix, with a row of zeroes on the bottom, but not an empty column for x₃, you would write x₃=r or what have you and then include r when solving for x₁ and x₂. In this case, where x₁ and x₂ simply equal zero, what would one write about x₃?

x₃ is arbitrary, meaning it can have any value.

 

[Karnage1993]

In this case, you still have to set $x_3 = r$ for some $r \in \mathbb{R}$.

 

[MoreDrinks]

x₃ is arbitrary, meaning it can have any value.

In this case, you still have to set $x_3 = r$ for some $r \in \mathbb{R}$.

Thank you both.