$27x + 6y - z = 85$

Hence  $27x = 85 -6y + z$

$x= \dfrac{ 1}{ 27} (85 - 6y + z)\cdots \mathrm{Equation \ 1}$

$6x + 15y + 2z = 72 $

$15 y = 72 -6x -2z$

$y= \dfrac{ 1}{ 15} (72 - 6x - 2x)\cdots \mathrm{Equation \ 2 }$

$x + y + 54z =110$

$54z= 110 -x - y$

$z= \dfrac{ 1}{54} (110 - x - y)\cdots \mathrm{Equation \ 3}$

First Iteration:

• $Put \ y=0,\ z=0\ in\ Equation\ (1)\ to\ find\ x_1, $

   $x_1= \dfrac {1}{ 27}(85 - 6y +z)$

$=\dfrac {1}{ 27}(85 - 0 +0)$

$x_1 = 3.15$

• ​$Put \ x=3.15,\ z=0\ in\ Equation\ (2)\ to\ find\ y_1, $

$y_1=\dfrac {1}{15} (72 -6x -2z)$

$=\dfrac {1}{15}[72 -6(3.15) -0]$

$y_1 = 3.54$

• $We\ use\ values\ of\ x_1\ and\ y_1\ to\ find\ z_1\ that\ is\ we\ put\ x=3.15; y=3.54\ in\ Equation\ (3),$

    $z_1=\dfrac {1} {54} (110 - x - y)$

   $=\dfrac {1} {54} (110 - 3.15 - 3.54)$

   $z_1 = 1.91$

   

Second Iteration:

• $We\ use\ values\ of\ y\ and\ z\ to\ find\ x\ that\ is\ we\ put\ y=3.54; z=1.91\ in\ Equation\ (1)\ to\ find\ x_2;$

$x_2= \dfrac {1}{27} (85 - 6y + z)$

= $\dfrac {1}{27} [85 - 6(3.54) + 1.91]$

$x_2 = 2.43$

• $Put \ x=2.43,\ z=1.91\ in\ Equation\ (2)\ to\ find\ y_2, $

$y_2=\dfrac {1} {15} (72 - 6x - 2z)$

$=\dfrac {1} {15} [72 - 6(2.43) - 2(1.91)]$

 $y_2 = 3.57$

• $Put \ x=2.43,\ y=3.57\ in\ Equation\ (3)\ to\ find\ z_2, $

$z_2 =\dfrac {1}{ 54} (110 - x - y)$

= $\dfrac {1}{ 54} (110 - 2.43 - 3.57)$

$z_2 = 1.93$

Third Iteration:

• $Put \ y=3.57,\ z=1.93\ in\ Equation\ (1)\ to\ find\ x_3, $

$x_3 = \dfrac {1}{27} (85 - 6y + z)$

$=\dfrac {1}{27} [85 - 6(3.57) + 1.93]$

$x_3 = 2.43$

• $Put \ x = 2.43,\ z=1.93\ in\ Equation\ (2)\ to\ find\ y_3,$

   $y_3 = \dfrac {1}{15} (72 - 6x - 2z)$

$= \dfrac {1}{15} [72 - 6(2.43) - 2(1.93)]$

$y_3 = 3.57$

• $Put \ x= 2.43,\ y=3.57\ in\ Equation\ (3)\ to\ find\ z_3,$

$z_3=\dfrac {1}{54} (110 - x - y)$

$=\dfrac {1}{54} (110 - 2.43 - 3.57)$

$z_3 = 1.93$

• $As\ the\ second\ and\ third\ iteration\ give\ same\ values,\ x = 2.43, y = 3.57, z=1.93$