How many scaler equations in Maxwell's equations?
Maxwell's equations contain two scalar equations and two vector equations. Gauss' law and Gauss' law for magnetism are the scalar equations. The Maxwell-Faraday equation and Ampere's circuital law are the vector equations.
A system of equations may have any amount of solutions. If the equations are linear, the system will have either no solution, one solution, or an infinite number of solutions. If the equations are linear AND there are as many equations as variables, AND they are independent, the system will have exactly one solution.