linear algebra

Question

Find the value(s) of the constant k such that the system of linear equations:

x + 2ky = k,

2kx + 4ky = 4k − 2

has

(i) No solution.

(ii) An infinite number of solutions.

(iii) Exactly one solution

Answer 1

Avi

Rearrange the equations giving expressions for y

y = (-x + k) / 2k = (-x/2k) + (1/2)

y = (-2kx + 4k-2) / 4k = (-1/2) x + (4k-2) / 4k

(i) No solution if lines parallel , and not identical

parallel if -1/2k = -1/2 ie k = 1. in this case lines will be identical. so no value of k giving no solution

(ii) infinite number of solutions, where lines identical, k = 1

(iii) one solution - k not equal to 1. here lines not parallel so must intersect at 1 point

Mike