Question

Examine the consistency of the system of equations:

x + y + z = 7, x + 2y + 3z = 18, y + 2z = 6

Answer 1

x + y + z = 7

x + 2y + 3z = 18

y + 2z = 6

The matrix form of the equations

`[(1, 1, 1),(1, 2, 3),(0, 1, 2)] [(x),(y),(z)] = [(7),(18),(6)]`
        A           X     =     B

Augmented Matrix [A, B]	Elementary Transformation
`[(1, 1, 1, 7),(1, 2, 3, 18),(0, 1, 2, 6)]`
`∼[(1, 1, 1, 7),(0, 1, 2, 11),(0, 1, 2, 6)]`	`{:"R"_2 -> "R"_2 - "R"_1:}`
`∼[(1, 1, 1, 7),(0, 1, 2, 11),(0, 0, 0, -5)]`	`{:"R"_3 -> "R"_3 - "R"_2:}`
p(A) = 2; p(A, B) = 3

Here p(A) ≠ p(A, B)

∴ The given system is inconsistent and has no solution.