There is a system of linear equations which is consistent and has a unique solution. If you change the numbers on the right hand sides of the equations ( and only those ), then can it happen system of equations
1) has no solutions
2) has infinitely many solutions
None of the two issues is possible.
Indeed, the "unique solution property" (sometimes called "Cramer") of a system $AX=B$ is attached to the matrix of the system (see remark), i.e., only to coefficients on the LHS. The composition of the RHS doesn't matter.
Remark: this condition is $\det(A) \neq 0$.