Question

Examine the consistency of the system
$x-3y-8z=-10,3x+y-4z=0,2x+5y+6z-13=0$ by using rank method and hence solved the system

Answer 1

The given system of equations is equivalent to the single matrix equation.

$\left[\begin{array}{ccc}1& -3& -8\\ 3& 1& -4\\ 2& 5& 6\end{array}\right]\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$ $=\left[\begin{array}{c}-10\\ 0\\ 13\end{array}\right]$
i.e., $AX=B$
The augumented martix is
$[A,B]=\left[\begin{array}{cccc}1& -3& -8& -10\\ 3& 1& -4& 0\\ 2& 5& 6& 13\end{array}\right]$
$\sim \left[\begin{array}{cccc}1& -3& -8& -10\\ 10& 10& 20& 30\\ 0& 11& 22& 33\end{array}\right]R_2\to R_2-3R_1{R}_3\to R_3-2R_1$
$-\left[\begin{array}{cccc}1& -3& -8& -10\\ 0& 1& 2& 3\\ 0& 1& 2& 3\end{array}\right]R_2\to \frac{1}{10}{R}_2{R}_3\to \frac{1}{11}{R}_3$
$-\left[\begin{array}{cccc}1& -3& -8& -10\\ 0& 1& 2& 3\\ 0& 0& 0& 0\end{array}\right]R_3\to R_3-R_2$
The last equivalent matrix is in the echelon form. It has two non-zero rows.
$\therefore \rho (A,B)=2$
Also $A\sim \left[\begin{array}{ccc}1& -3& -8\\ 0& 1& 2\\ 0& 0& 0\end{array}\right]$
Since there are two non-zero rows, $\rho \left(A\right)=2$
Here $\rho \left(A\right)=\rho [A,B]=2\ne$ number of unknowns.
$\therefore$ The given system is consistent but has infinitely many solution.

The given system is equivalent to the matrix equation
$x-3y-8z=-10....\left(1\right)$
$y+2z=3....\left(2\right)$
Assuming $z=k$,
$\left(1\right)\Rightarrow x-3y-8k-10\Rightarrow x-3y=-10+8k....\left(3\right)$
$\left(2\right)\Rightarrow y+2k=3\Rightarrow y=3-2k....\left(4\right)$
$\left(3\right)\Rightarrow x-3y=-10+8k$
Substituting $y=3-2k$
$x-3(3-2k)=-10+8k$
$\Rightarrow x-9+6k=-10+8k$
$\Rightarrow x=8k-6k-10+9$
$\Rightarrow x=2k-1$
$\therefore$ The solution is $(2k-1,-2k+3,k)$, where $k\in R$