Usually with a system of three linear equations and 3 variables, there will be one solution. But sometimes there are inconsistencies in the equations and no solutions are possible. Sometimes two of the three equations, or all of them, are effectively the same and then there will be many solutions. All that can be found is a relationship between the variables.
For a non-linear system, there can be a finite number of solutions for three variables. For example, if a quadratic equation is involved, one variable may have two possible solutions. This may affect the number of solutions for the other variables.