The-system-of-linear-equations-x-y-z-2-2x-y-z-3-3x-2y-kz-4-has-a-unique-solution-if-

Question Number 81876 by zainal tanjung last updated on 16/Feb/20

The system of linear equations x + y + z = 2,

2 x + y - z = 3, 3 x + 2 y + k z = 4 has a unique

solution if

Commented by Tony Lin last updated on 16/Feb/20

Δ = | \begin{matrix} 1 & 1 & 1 \\ 2 & 1 & - 1 \\ 3 & 2 & k \end{matrix} |

= k - 3 + 4 - 3 + 2 - 2 k

= - k \neq 0

\Rightarrow k \neq 0

Δ_{x} = | \begin{matrix} 2 & 1 & 1 \\ 3 & 1 & - 1 \\ 4 & 2 & 0 \end{matrix} |

= 6 - 4 - 4 + 4 = 2 \neq 0

\Rightarrow w h e n k = 0, t h e e q u a t i o n h a s n o s o l u t i o n

k \neq 0, t h e e q u a t i o n h a s a u n i q u e s o l u t i o n