How does the number of solutions of a system depend on ranks of the coefficient matrix and the augmented matrix?

The number of solutions of a system is given by the number of different variables in the system, this number has to be the same as the number of independent equations. The coefficients and the augmented matrix of the system show these values in a matrix form. A system has a unique solution when the rank of both matrixes and the number o variables in the system are the same.

Step-by-step explanation:

For example, the following system has 2 different variables, x and y.

In order to find a unique solution to the system, the number of independent equations and variables in the system must be the same In the previous example, you have 2 independent equations and 2 variables, then the solution of the system is unique.  

The rank of a matrix is the dimension of the vector generated by the columns, in other words, the rank is the number of independent columns of the matrix.  

According to Rouché-Capelli Theorem, a system of equations is inconsistent if the rank of the augmented matrix is greater than the rank of the coefficient matrix. The inconsistency of the system is because you can't find a combination of the variables that will solve the system.