The values of b and z determine the solvability of the system of equations; a unique determinant for unique solutions, consistency with zero determinant for multiple solutions, and inconsistency for no solutions.
To determine the value of b and z for a system of equations to have no solution, more than one solution, or a unique solution, we use Gauss elimination method. For a unique solution, the system must have three pivots - one for each variable, ensuring the determinant is not zero.
If the determinant is zero and the equations are inconsistent, the system has no solution; if the determinant is zero and the system is consistent, we have more than one solution. In general, b should not make any row a multiple of another, and z is dependent on other coefficients which are not specified in the system given.