Past exam of the mathematical methods I course of the University of Oxford / 2014 resit / 8 / Solution

Subtracting the first equation from the third gives

x - z = 3/2

. The second equation then becomes

(α - 2) x = β - 3

, and the third fixes

y = (7 - 7 x) /5

. The linear system has the following three consistency cases:

α \neq = 2 : α = 2, β = 3 : α = 2, β \neq = 3 : (x, y, z) = (\frac{β - 3}{α - 2}, \frac{7 ( α - β + 1 )}{5 ( α - 2 )}, \frac{2 β - 3 α}{2 ( α - 2 )}); (x, y, z) = (t, \frac{7 ( 1 - t )}{5}, t - \frac{3}{2}), t \in R; no solution .

The first case is a unique solution, the second is an affine line in a vector space of solutions, and the third is an inconsistent linear system.

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