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in Applications of Matrices and Determinants by (46.9k points)
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Find k if the equations 2x + 3y – z = 5; 3x – y + 4z = 2; x + 7y – 6z = k are consistent.

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Given that the system 2x + 3y – z = 5; 3x – y + 4z = 2, x + 7y – 6z = k is consistent.

ρ(A) = 2 ρ([A, B]) = 2 or 3 

For the equations to be consistent, ρ([A, B]) = ρ(A) = 2 

-8 + k = 0 ⇒ k = 8

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