# Examine the consistency of the following system of equation. If it is consistent than solve the same. $x-3y-8z=-10\;,3x+y-4z=0\;,2x+5y+6z-13=0$

Toolbox:
• Rank method for finding consistency of a system of $m$ equation in $n$ unknowns : (i) Write the matrix equation $AX = B$ (ii) Find the angmented matrix $[ A, B]$ (iii) Find the ranks of A and [A,B] by elementary row transformations. (iv) (a) $\rho(A) \neq \rho[A,B] \Rightarrow$ inconsistency with no solution (b) $\rho(A) = \rho[A,B] = n$ where $n$ is the number of unknowns $\Rightarrow$ consistency with unique solution. (c) If $\rho(A)=\rho[A,B] < n,$ the system is consistent with infinite number of solutions. (d) A system of homogeneous linear equations is always consistent, if $\rho(A) = n$ then the only solution is the trivial solution. If $\rho(A) < n$ then the system has nontrivial solutions in addition to the trivial solutions.
Step 1
The matrix equation is $\begin{bmatrix} 1 & -3 & -8 \\ 3 & 1 & -4 \\ 2 & 5 & 6 \end{bmatrix} \begin{bmatrix}x \\ y \\ z \end{bmatrix} = \begin{bmatrix}-10 \\ 0 \\ 13 \end{bmatrix}$
$AX = B$
The angmented matrix
$[A,B] = \begin{bmatrix}1 & -3 & -8 & -10 \\ 3 & 1 & -4 & 0 \\ 2 & 5 & 6 & 13 \end{bmatrix}$
$\sim \begin{bmatrix}1 & -3 & -8 & -10 \\ 0 & 10 & 20 & 30 \\ 0 & 11 & 22 & 33 \end{bmatrix} R_2 \rightarrow R_2-3R_1$
$R_3 \rightarrow R_3-2R_1$
$\sim \begin{bmatrix}1 & -3 & -8 & -10 \\ 0 & 10 & 20 & 30 \\ 0 & 0 & 0 & 0 \end{bmatrix} R_2 \rightarrow R_3 - \large\frac{11}{10}R_2$
Step 2
There are two rows of nonzero elements in the last matrix. It can be seen that $\rho(A) = \rho[A,B]=2$
The equation are consistent with infinite number of solutions.
Step 3
They reduce to
$x-3y-8z=-10$
$\: \: \: \: \: \: \: \: \: y+2z=3 \: Let \: z=k_1\: \: k \in R$
Then $y = 3-2k\: and \:x=-10+3(3-2k)+8k$
$= 2k-1$
$(x,y,z) = (2k-1, \: 3-2k, \: k)\: k \in R$

edited Jun 3, 2013