Answer
$\{(1,3,2)\}$.
Work Step by Step
The given system of equations is
$\left\{\begin{matrix}
x& +y &+z&=&6& ...... (1) \\
3x& +4y & -7z&=&1& ...... (2)\\
2x& -y &+3z &=&5& ...... (3)
\end{matrix}\right.$
Addition method:-
Step 1:- Reduce the system to two equations in two variables.
Multiply the equation (1) by $-4$.
$\Rightarrow -4x-4y -4z=-24 $ ...... (4)
Add equation (2) and (4).
$\Rightarrow 3x+4y-7z-4x-4y -4z=1-24 $
Simplify.
$\Rightarrow -x-11z =-23 $ ...... (5)
Add equation (1) and (3).
$\Rightarrow x+y+z+2x-y +3z=6+5 $
$\Rightarrow 3x+4z=11 $ ...... (6)
Step 2:- Solve the two equations from the step 1.
Multiply equation (5) by $3$.
$\Rightarrow -3x-33z =-69 $ ...... (7)
Add equation (6) and (7).
$\Rightarrow 3x+4z-3x-33z=11-69 $
Add like terms.
$\Rightarrow -29z=-58 $
Divide both sides by $-29$.
$\Rightarrow \frac{-29z}{-29}=\frac{-58}{-29} $
$\Rightarrow z=2$
Substitute the value of $z$ into equation (6).
$\Rightarrow 3x+4(2) =11 $
$\Rightarrow 3x+8 =11 $
Subtract $8$ from both sides.
$\Rightarrow 3x+8-8 =11-8 $
Simplify and isolate $x$.
$\Rightarrow x =1 $
Step 3:- Use back-substitution to find the remaining variables.
Substitute the value of $x$ and $z$ into equation (1).
$\Rightarrow 1 +y +2=6$
Simplify.
$\Rightarrow y +3=6$
Subtract $3$ from both sides.
$\Rightarrow y +3-3=6-3$
Add like terms.
$\Rightarrow y =3$
The solution set is $\{(x,y,z)\}=\{(1,3,2)\}$.
