Answer
$(1,3,2)$
Work Step by Step
We are given the system:
$\begin{cases}
x+y+z=6\\
3x+4y-7z=1\\
2x-y+3z=5
\end{cases}$
We will use the addition method. Multiply Equation 1 by -4 and add it to Equation 2 to eliminate $y$. Also add Equation 1 to Equation 3 to eliminate $y$:
$\begin{cases}
3x+4y-7z-4(x+y+z)=1-4(6)\\
2x-y+3z+x+y+z=5+6
\end{cases}$
$\begin{cases}
3x+4y-7z-4x-4y-4z=-23\\
2x-y+3z+x+y+z=11
\end{cases}$
$\begin{cases}
-x-11z=-23\\
3x+4z=11
\end{cases}$
Multiply Equation 1 by 3 and add it to Equation 2 to eliminate $x$ and determine $z$:
$\begin{cases}
3(-x-11z)=3(-23)\\
3x+4z=11
\end{cases}$
$\begin{cases}
-3x-33z=-69\\
3x+4z=11
\end{cases}$
$-3x-33z+3x+4z=-69+11$
$-29z=-58$
$z=2$
Substitute the value of $z$ in the Equation $-x-11z=-23$ to determine $z$:
$-x-11(2)=-23$
$-x-22=-23$
$x=1$
Substitute the values of $x, z$ is Equation 1 of the given system to find $y$:
$x+y+z=6$
$1+y+2=6$
$y+3=6$
$y=3$
The system's solution is:
$(1,3,2)$
