Question

Examine the consistency of the system of equations $4x+3y+6z=25;x+5y+7z=13;2x+9y+z=1$ by using rank method. If it is consistent. solve them.

Answer 1

Let the matrix $A$ be $\left[\begin{array}{ccc}4& 3& 6\\ 1& 5& 7\\ 2& 9& 1\end{array}\right]$ and matrix $B$ be $\left[\begin{array}{c}25\\ 13\\ 1\end{array}\right]$

$|A|\ne 0$ , therefore the rank of $A$ is $3$

Now consider $A|B=\left[\begin{array}{cccc}4& 3& 6& 25\\ 1& 5& 7& 13\\ 2& 9& 1& 1\end{array}\right]$ , the rank is $3$

Therefore $\rho \left(A\right)=\rho \left(A|B\right)=3$ , which implies that the system of equations is consistent.

$\rho \left(A\right)=\rho \left(A|B\right)=$ number of rows , therefore the sytem has unique solution.

We will solve it by determinant method.

The value of $D=\left|\begin{array}{ccc}4& 3& 6\\ 1& 5& 7\\ 2& 9& 1\end{array}\right|=-199$

$D_x=\left|\begin{array}{ccc}3& 6& 25\\ 5& 7& 13\\ 9& 1& 1\end{array}\right|=-796$

$D_y=\left|\begin{array}{ccc}4& 1& 2\\ 25& 13& 1\\ 6& 7& 1\end{array}\right|=199$ ,

$D_z=\left|\begin{array}{ccc}4& 1& 2\\ 3& 5& 9\\ 25& 13& 1\end{array}\right|=-398$

Therefore $x=\frac{-796}{-199}=4$ , $y=\frac{199}{-199}=-1$ and $z=\frac{-398}{-199}=2$

Hence $4,-1,2$ is the solution of the given simultaneous equations.