 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2017-resit/5/a/solution

x = 2 y - 1

and

z = 3 - 4 y

. The third then reduces to

(α - 2) y = β - 3

. Hence

α = 2, β \neq = 3 : α = 2, β = 3 : α \neq = 2 : no solution, infinitely many solutions, one solution .

In the infinite family

y = t

x = 2 t - 1

and

z = 3 - 4 t

, for

t \in R

. This makes the distinction between an inconsistent linear system and a free parameter explicit; its coefficient matrix has determinant

4 - 2 α

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