Answer
The solution of the system of equations is $x=7,y=4\text{ and }z=5$.
Work Step by Step
Let us consider the given equation:
$x+y+z=16$ …… (I)
$2x+3y+4z=46$ …… (II)
$5x-y=31$ …… (III)\
Equation (I): multiply by $4$ and substrate from equation (II):
$\begin{align}
& 2x+3y+4z=46 \\
& 4x+4y+4z=64 \\
& \underline{-\text{ }-\text{ }-\text{ }-} \\
& -2x-y=-18 \\
\end{align}$
Further simpliy:
$2x+y=18$ …… (IV)
By adding equation (III) and equation (IV): we get,
$\begin{align}
& 5x-y=31 \\
& \underline{2x+y=18} \\
& 7x=49 \\
& \text{ }x=7 \\
\end{align}$
Putting in the value of $x=7$ in equation (III). We get,
$\begin{align}
& 35-y=31 \\
& -y=-4 \\
& y=4
\end{align}$
Substitute $x=7$ and $y=4$ in equation (I):
$\begin{align}
& 7+4+z=16 \\
& z=5
\end{align}$
Thus, the solution of the system of equations is $x=7,y=4\text{ and }z=5$.
