Question

prove system of linear equations has a unique solution

Answer 1

This is when there are more equations than variables. Assuming you have m variables and n equations, there will only be one solution if some n-m equations are be linearly dependent on the other m equations in the system. The example you have falls in this category. One way to test if a solution exists is to remove the linearly dependant equations, and take the determinant of the the remaining system. If the determinant is non-zero of this reduced system, a solution exists.

For example
3x = 0
x = 1
This system has no solution.

Whereas
3x = 0
x = 0
has a unique solution

Underdetermined
This is when you have more variables than equations. In this case there can be many solutions that satisfy the system.

Well determined
This is when you have equal number of variables and equations. There will be only one solution if all the equations are linearly independent. There can be, at most, only one solution.