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Question
Find k if the equations 2x + 3y – z = 5, 3x – y + 4z = 2, x + 7y – 6z = k are consistent
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Solution
2x + 3y – z = 5
3x – y + 4z = 2
x + 7y – 6z = k
The matrix form of these equations
`[(2, 3, -1),(3, -1, 4),(1, 7, -6)] [(x),(y),(z)] = [(5),(2),("k")]`
A X = B
| Augmented Matrix [A, B] |
Elementary Transformation |
| `[(2, 3, -1, 5),(3, -1, 4, 2),(1, 7, -6, "k")]` | |
| `[(1, 7, -6, "k"),(3, -1, 4, 2),(2, 3, -1, 5)]` | `{:"R"_1 ↔ "R"_3:}` |
| `[(1, 7, -6, "k"),(0, -22, 22, 2 - 3"k"),(0, -11, 11, 5 - 2"k")]` | `{:("R"_2 -> "R"_2 - 3"R"_1),("R"_3 -> "R"_3 - 2"R"_1):}` |
| `[(1, 7, -6, "k"),(0, -11, 11, 5 - 2"k"),(0, -22, 22, 2 - 3"k")]` | `{:"R"_2 ↔ "R"_3:}` |
| `[(1, 7, -6, "k"),(0, -11, 11, 5 - 2"k"),(0, 0, 0, "k" - 8)]` | `{:"R"_3 -> "R"_3 - 2"R"_2:}` |
| p(A) = 2; p(A, B) = 2 if k = 8 |
Obviously, the last equivalent matrix is in the ech-elon form.
Since the equations are consistent
p(– A) = p(A, B)
p(A) = 2 and p(A, B) = 2 then
k = 8
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