# Solve the following non-homogeneous system of linear equation by determinant method: $\large \frac{1}{x}+\frac{2}{y}-\frac{1}{z}=1\;;\frac{2}{x}+\frac{4}{y}+\frac{1}{z}=5\;;\frac{3}{x}-\frac{2}{y}-\frac{2}{z}=0$

Toolbox:
• Consistency of a system of 2 ( or 3 ) linear equations in 2 ( or 3 ) unknowns. Case 1 : If $\Delta \neq 0$ thesystem is consistent and has a unique solution. Cramer's rule can be used. Case 2 : If $\Delta = 0$, there are two / ( four) possibilities:
• 2a: If $\Delta = 0$, and atleast one of the values $\Delta_ x, \Delta_ y ( or \: \Delta_ z )$ is nonzero, the system is inconsistent and has no solution.
• b: If $\Delta = 0\: and \: \Delta_x=\Delta_y ( = \Delta_z)=0$ and at least one element ( second order minor ) of $\Delta$ is nonzero, the system is consistent with infinitely many solution. The system reduces to one equation ( two equations )
• 2c: If $\Delta = 0$ and $\Delta_x=\Delta_y = \Delta_z=0$ and all their 2 x 2 minors are 0, and $\Delta$ has atleast one nonzero element, then the system is consistent with infinitely many solutions. The system reduces to one equation.
• 2d: If $\Delta = 0, \Delta_x=\Delta_y = \Delta_z=0$, all 2 x 2 minors of $\Delta$ are zero but there is at least one 2 x 2 minor of $\Delta_x, \Delta_y \: or \: \Delta_z$ that is nonzero, then the system is inconsistent and has no solution.
Step 1
Let $u = \large\frac{1}{x}, v = \large\frac{1}{y}, w = \large\frac{1}{z}$
The equations reduce to
$u+2v-w=1$
$2u+4v+w=5$
$3u-2v-2w=0$
$\Delta = \begin{vmatrix} 1 & 2 & -1 \\ 2 & 4 & +1 \\ 3 & -2 & -2 \end{vmatrix} = 1(-8+2)-2(-4-3)-1(-4-12)$
$= -6+14+16=24 \neq 0$
$\therefore$ Cramer's rule can be used.
Step 2
$\Delta_u = \begin{vmatrix} 1 & 2 & -1 \\ 5 & 4 & 1 \\ 0 & -2 & -2 \end{vmatrix} = 1(-8+2)-5(-4-2)$
$-6+30=24$
$\Delta_v = \begin{vmatrix} 1 & 1 & -1 \\ 2 & 5 & 1 \\ 3 & 0 & -2 \end{vmatrix} = 1(-10-0)-1(-4-3)-1(0-15)$
$-10+7+15=12$
$\Delta_w = \begin{vmatrix} 1 & 2 & 1 \\ 2 & 4 & 5 \\ 3 & -2 & 0 \end{vmatrix} = 1(0+10)-2(0-15)+1(-4-12)$
$=10+30-16=24$
Step 3
$u = \large\frac{\Delta_u}{\Delta} = \large\frac{24}{24}=1,\: \: v = \large\frac{\Delta_v}{\Delta}=\large\frac{12}{24}=\large\frac{1}{2}, \: \: w = \large\frac{\Delta_w}{\Delta}=\large\frac{24}{24}=1$
$x = \large\frac{1}{u}=1,\: \: y = \large\frac{1}{v}=1, \: \: z = \large\frac{1}{w}=1$
$(x, y, z) = (1, 2, 1)$

edited Jun 3, 2013