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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

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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 $\det(A) \neq 0$.

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