Question

Find k if the equations x + y + z = 1, 3x – y – z = 4, x + 5y + 5z = k are inconsistent

Answer 1

x + y + z = 1

3x – y – z = 4

x + 5y + 5z = k

The matrix equation corresponding to the given system is

`[(1, 1, 1),(3, -1, -1),(1, 5, 5)] [(x),(y),(z)] = [(1), (4), ("k")]`
          A                X    =     B

Augmented Matrix [A, B]	Elementary Transformation
`[(1, 1, 1, 1),(3, -1, -1, 4),(1, 5, 5, "k")]`
`∼[(1, 1, 1, 1),(0, -4, -4, 1),(0, 4, 4, "k" - 1)]`	`{:("R"_2 -> "R"_2 - 3"R"_1),("R"_3 -> "R"_3 - "R"_1):}`
`∼[(1, 1, 1, 1),(0, -4, -4, 1),(0, 0, 0, "k")]`	`{:"R"_3 -> "R"_3 + R_2:}`

Obviously, the last equivalent matrix is in the echelon form.

Since the equations are inconsistent

p(A) ≠ p(A, B)

Here p(A) = 2 but p(A, B) should not equal to 2

∴ k ≠ 0

The equations are inconsistent when k assume any real value other than 0.