# Suppose that a nonhomogeneous system with 10 linear equations in 8 unknowns has a solution with 2 free variables. Is it possible to change some constants on the equations’ right hand side to make the new system inconsistent?

Suppose that a nonhomogeneous system with 10 linear equations in 8 unknowns has a solution with 2 free variables. Is it possible to change some constants on the equations’ right hand side to make the new system inconsistent?
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Nathaniel Kramer
Step 1
The system is equivalent to Ax=b.
We have 10 linear equations in 8 unknowns. So dimension of $A=10×8$.
Step 2
Ax=b has a solution with 2 free variables. So, dim Nul A=2.
So, dimension of column= 8-2=6.
So, $x\to Ax$ is onto R6. Hence, every right hand side is obtainable.
Result: No, it is not possible to change some constants on the equations’ right hand side to make the new system inconsistent.