Question

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

• 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 $A =\left[\begin{matrix}3&-1&1\\ 2&-2&3\\ 1&1&-1\end{matrix}\right]$, $X=\left[\begin{matrix}x\\ y\\ z\end{matrix}\right]$ and $B =\left[\begin{matrix}5\\ 7\\ -1\end{matrix}\right]$

Now, $\left|A\right|=3\left(2-3\right)-\left(-1\right)\left(-2-3\right)+1\left(2+2\right)=-4 \ne0$
$\Rightarrow \quad A^{-1}$ exists and so the given system has a unique solution $X=A^{-1}B$
Now, $adj A =\left[\begin{matrix}-1&0&-1\\ 5&-4&-7\\ 4&-4&-4\end{matrix}\right]$
$\therefore \quad$ $A^{-1}=\frac{1}{\left|A\right|} adj \,A=-\frac{1}{4}\left[\begin{matrix}-1&0&-1\\ 5&-4&-7\\ 4&-4&-4\end{matrix}\right]$

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

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

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