Question

Use Cramer's rule to find the solution of the following systems of linear equation in terms of the parameter K.

a) 5x - ky = 6. b) 2x - 3y = k

- 2x + 2ky = -3 X + 2y = - 2

Answer 1

a)

$\begin{cases} 5x-ky = 6 \\ -2x+2ky = -3 \end{cases} \\ \Delta =\begin{vmatrix} 5 & -k \\ -2 & 2k \end{vmatrix} =10k -2k = 8k \\ \Delta_1 = \begin{vmatrix} 6 &-k\\ 3&2 \end{vmatrix} = 12k-3k = 9k\\ \Delta_2 = \begin{vmatrix} 5&6\\ -2&-3 \end{vmatrix} = -15+12 = -3 \\ x = \cfrac{\Delta_1}{\Delta} = \cfrac{9k}{8k} = \cfrac{9}{8}\\ y = \cfrac{\Delta_2}{\Delta} =\cfrac{-3}{8k}$

b)

$\begin{cases} 2x - 3y = k\\ x+2y = -2 \end{cases}\\ \Delta = \begin{vmatrix} 2 & -3 \\ 1 & 2 \end{vmatrix} = 7\\ \Delta_1 = \begin{vmatrix} k&-3\\ -2&2 \end{vmatrix} =2k-6\\ \Delta_2= \begin{vmatrix} 2&k\\ 1&-2 \end{vmatrix} = -4-k\\ x = \cfrac{\Delta_1}{\Delta} =\cfrac{2k-6}{7}\\ y = \cfrac{\Delta_2}{\Delta} = -\cfrac{4+k}{7}$