Answer
The solution of the system is $\left( 2,-1,1 \right)$.
Work Step by Step
Let us consider the given equation:
$4x-y+2z=11$ …… (I)
$x+2y-z=-1$ …… (II)
$2x+2y-3z=-1$ …… (III)
$2\times $ equation (II), add equation (1)
$\begin{align}
& 4x-y+2z=11 \\
& 2x+4y-2z=-2 \\
& \overline{6x+3y=9\,\,\,\,\,\,\,\,\,\,\,\,\,\,} \\
\end{align}$
$2x+y=3$ …… (IV)
Equation (III) subtract $3\times $ Equation (II):
$\begin{align}
& 3x+6y-3z=-3 \\
& 2x+2y-3z=-1 \\
& -\text{ }-\text{ }+\text{ }+ \\
& \overline{\,\,\,\,\,\,\,x+4y=-2\,\,\,\,\,\,} \\
\end{align}$
$x+4y=-2$ …… (V)
Now equation (IV) subtract $2\times $ equation (V):
$\begin{align}
& 2x+y=3 \\
& 2x+8y=-4 \\
& -\text{ }-\text{ }+ \\
& \overline{\begin{align}
& \,\,\,\,\,\,\,-7y=7\,\,\,\,\,\, \\
& y=-1 \\
\end{align}} \\
\end{align}$
Now, putting the value of $y=-1$ in equation (IV):
$\begin{align}
& 2x-1=3 \\
& 2x=4 \\
& x=2
\end{align}$
And, putting the value of x and y in equation (I):
$\begin{align}
& 4\left( 2 \right)-\left( -1 \right)+2z=11 \\
& 8+1+2z=11 \\
& 2z=2 \\
& z=1
\end{align}$
Thus, the order triple $\left( 2,-1,1 \right)$ satisfies the three equations.
