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# Examine the consistency of the following system of equation. If it is consistent than solve the same. $4x+3y+6z=25\;,x+5y+7z=13\;,2x+9y+z=1$

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 $\begin{bmatrix} 4 & 3 & 6 \\ 1 & 5 & 7 \\ 2 & 9 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 25 \\ 13 \\ 1 \end{bmatrix}$
$AX=B$
Angmented matrix $[A, B] = \begin{bmatrix} 4 & 3 & 6 & 25 \\ 1 & 5 & 7 & 13 \\ 2 & 9 & 1 & 1 \end{bmatrix}$
$[A,B] \sim \begin{bmatrix} 1 & 5 & 7 & 13 \\ 4 & 3 & 6 & 25 \\ 2 & 9 & 1 & 1 \end{bmatrix} R_1 \leftrightarrow R_2$
$\sim \begin{bmatrix} 1 & 5 & 7 & 13 \\ 0 & -17 & -22 & -27 \\ 0 & -1 & -13 & -25 \end{bmatrix} R_2 \rightarrow R_2 - 4R_1$
$R_3 \rightarrow R_3-4R_1$
$\sim \begin{bmatrix} 1 & 5 & 7 & 13 \\ 0 & 1 & 13 & 25 \\ 0 & 17 & 22 & 27 \end{bmatrix}R_2 \leftrightarrow R_3$
$\sim \begin{bmatrix} 1 & 5 & 7 & 13 \\ 0 & 1 & 13 & 25 \\ 0 & 0 & 199 & 398 \end{bmatrix}R_3 \rightarrow R_3-17R_2$
Step 2
$\rho(A) = \rho(A,B) = 3$ as there are no nonzero rows. The equations are consistent and have a unique solution.
Step 3
They reduce to
$x+5y+7z=13$
$\: \: \: \: \: \: \: \: \: \: y+13z=25$
$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: 199z = 398$
We have $z = 2,\: y=25-13(2)=-1$
$x = 13-5(-1)-7(2)$
$= 13+5-14=4$
$(x,y,z) = (4, -1, 2)$

edited Jun 3, 2013