Answer
Refer to the description below.
Work Step by Step
There isn't a single solution for each right-hand side of a nonhomogeneous system of seven equations with six unknowns. This is due to the fact that a nonhomogeneous system of equations can only have a single solution if the coefficient matrix's rank is the same as the augmented matrix's rank (which includes the constants on the right-hand side). The coefficient matrix only has a maximum rank of 6, whereas the augmented matrix only has a maximum rank of 7. This is because there are more equations than unknowns in this situation. There will be certain right-hand sides for which the system has no solution, infinitely many solutions, or a single solution, hence the ranks cannot be equal for all possible right-hand sides.
However, it is possible for a nonhomogeneous system of seven equations in six unknowns to have a unique solution for some right-hand side of constants. This can happen if the coefficient matrix happens to have full rank (i.e., rank 6) and the right-hand side of the system is chosen so that the resulting augmented matrix also has rank 6. In this case, the system will have a unique solution, because there are exactly six linearly independent equations and six unknowns, so the solution can be uniquely determined. However, this is a special case and does not hold for all possible right-hand sides.
