# x+2y+z=5 To calculate: The solution for the system of equation

$x+2y+z=5$ To calculate: The solution for the system of equation $x+y-z=1$
$4x+7y+2z=16$
and if the system has one unique solution, write the solution set. Otherwise, determine the number of solutions to the system, and determine whether the system is inconsistent, or the equations are dependent.
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Calculation:
Consider the provided system of equations is,
$x+2y+z=5$......(1)
$x+y-z=1$......(2)
$4x+7y+2z=16$......(3)
Now, add both the equations (1) and (2).
$x+x+2y+y+z-z=5+1$
......(3) $2x+3y=6$
Consider the second and third equation from (1),
$x+y-z=1$
......(4) $4x+7y+2z=16$
Multiply the equation (1) by 2 and add with equation (3).
$2x+4x+2y+7y-2z+2z=2+16$
......(7) $6x+9y=18$
Multiply the equation (3) with —3 and with equation (7).
$-6x+6x+9y—9y=-18+18$
$0=0$
The system reduces to the identity $0=0$.
So, the system has infinitely many solutions.
$6x=24—5y$ Therefore, the system of equations $14=7z—3y$ has infinitely many
$4x—3z=10$ solutions.