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?

Cheyanne Leigh

Cheyanne Leigh

Answered question

2020-11-08

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?

Answer & Explanation

Nathaniel Kramer

Nathaniel Kramer

Skilled2020-11-09Added 78 answers

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, xAx 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.

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