Question

Solve the following system of equations :
$3x-y+z=5$, $2x-2y+3z=7$, $x+y-z=-1$

• A. $x=-1$ , $y=-1$ , $z=1$
• B. $x=1$ , $y=1$, $z=1$
• C. $x=1$, $y=-1$, $z=1$
• D. $x=1$, $y=-1$, $z=-1$

Answer 1

Answer: (C)

The given system of equations can be written as $AX=B$
where $A=\left[\begin{array}{ccc}3& -1& 1\\ 2& -2& 3\\ 1& 1& -1\end{array}\right]$, $X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$ and $B=\left[\begin{array}{c}5\\ 7\\ -1\end{array}\right]$

Now, $\left|A\right|=3\left(2-3\right)-\left(-1\right)\left(-2-3\right)+1\left(2+2\right)=-4\ne 0$
$\Rightarrow A^{-1}$ exists and so the given system has a unique solution $X=A^{-1}B$
Now, $adjA=\left[\begin{array}{ccc}-1& 0& -1\\ 5& -4& -7\\ 4& -4& -4\end{array}\right]$
$\therefore$ $A^{-1}=\frac{1}{|A|}adjA=-\frac{1}{4}\left[\begin{array}{ccc}-1& 0& -1\\ 5& -4& -7\\ 4& -4& -4\end{array}\right]$

$\therefore X=A^{-1}B=-\frac{1}{4}\left[\begin{array}{ccc}-1& 0& -1\\ 5& -4& -7\\ 4& -4& -4\end{array}\right]\left[\begin{array}{c}5\\ 7\\ -1\end{array}\right]=\left[\begin{array}{c}1\\ -1\\ 1\end{array}\right]$

$\Rightarrow \left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}1\\ -1\\ 1\end{array}\right]\Rightarrow x=1,y=-1,z=1$.

Hence, the solution of the given system of equations is
$x=1$, $y=-1$, $z=1$.