if you have an experimental setup which has three unknowns, how many linearly independent equations do you need to determine the unknowns to find one unique solution?

three equations

Explanation:

You need the same amount of linearly independent equations as the number of unknowns to find a unique solution. In this case where unknowns are three, you will need three linearly independent equations.

In order to check whether a system of linear equations has a unique solution or not we find out the determinant of the system. A non-zero determinant ensures that the system has a unique solution.