Question

Examine whether the following system is consistent or inconsistent and if consistent find the complete solution
$x+y+z=4;2x+5y-2z=3;x+7y-7z=5$

Answer 1

Given system of linear equations:

$x+y+z=4$

$2x+5y-2z=3$

$x+7y-7z=5$

Lets form in Matrix format,

$A=$ $\left[\begin{array}{ccc}1& 1& 1\\ 2& 5& -2\\ 1& 7& -7\end{array}\right]$

$B=$ $\left[\begin{array}{c}4\\ 3\\ 5\end{array}\right]$

$X=$ $\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$

Now for solving these equations, in matrix inverse method, apply the rule

$X=A^{-1}B$

$\Rightarrow$ $\text{DetA}=$ $\left|\begin{array}{ccc}1& 1& 1\\ 2& 5& -2\\ 1& 7& -7\end{array}\right|$

$\Rightarrow \text{DetA}=$ $1(-35+14)-1(-14+2)+1(14-5)$

$\Rightarrow \text{DetA}=$ $-21+12+9$

$\Rightarrow \text{DetA}=0$

$\Rightarrow A^{-1}$ does not exist.

Hence, The given system of linear equations are inconsistent.

i.e., Given system of linear equations have no solution.