Answer
The solution of the equation is $\left\{ \left( 8,-2,-2 \right) \right\}$.
Work Step by Step
The system of equations is given below:
$\left\{ \begin{align}
& x+2y+3z=-2 \\
& 3x+3y+10z=-2 \\
& 2y-5z=6
\end{align} \right.$
And the equation $2y-5z=6$ has only two variables x and y:
Now, consider the equations $ x+2y+3z=-2$ and $3x+3y+10z=-2$; eliminate $ z $ and obtain the equation containing the y and z variables in equation $2y-5z=6$.
Multiply the equation $ x+2y+3z=-2$ by 3 and equation $3x+3y+10z=-2$ by 1.
Therefore,
$\begin{align}
& \left( x+2y+3z \right)\times 3=-2\times 3 \\
& 3x+6y+9z=-6
\end{align}$
And
$3x+3y+10z=-2$
And subtract the equation $3x+6y+9z=-6$ from $3x+3y+10z=-2$ to obtain:
$\begin{align}
& \left( 3x+6y+9z \right)-\left( 3x+3y+10z \right)=\left( -6 \right)-\left( -2 \right) \\
& 3y-z=-4
\end{align}$
Now, solve the equations $2y-5z=6$ and $3y-z=-4$ ; the values of $ z $ and $ y $ are
$\begin{align}
& z=-2 \\
& y=-2
\end{align}$
Put values of z and y in equation $ x+2y+3z=-2$, to obtain the value of $ x $ as given below:
$\begin{align}
& x+2y+3z=-2 \\
& x+2\left( -2 \right)+3\left( -2 \right)=-2 \\
& x-4-6=-2 \\
& x=8
\end{align}$
Thus, the values of x, y, z are $\left\{ \left( 8,-2,-2 \right) \right\}$.
