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Question
Examine the consistency of the system of equations:
x + y + z = 7, x + 2y + 3z = 18, y + 2z = 6
Solution
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.
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