It is possible to estimate the solution to a linear system using an iterative process known as the GaussSeidel iteration method. Consider the linear system below:

$2x+3y=5$

$6x-2y=-7$

This can be rewritten in the form below.

$y=\frac{5-2x}{3}$

$x=\frac{-7+2y}{6}$

Note that the largest coefficient in each row has been used as the divisor.

Once this has been done, suitable starting estimates are chosen for $x$ and $y,$ often $0.$ The current values of $x$ and $y$ are then repeatedly passed into these equations to generate new values of $x$ and $y.$ After many iterations this often leads to a good approximation to the system.

We wish to solve the following linear system using Gauss$-$Seidel iteration.

$4x+2y+z=10$

$2x-8y+3z=4$

$x+2y+5z=6$

$($a$)$ Rearrange the first equation to give an equation for $x.$ Similarly rearrange the second to give you an equation for $y,$ and the third to give an equation for $z.$

$($b$)$ Using an initial estimate of $(0,0,0),$ iterate this set of equations $80$ times, replacing the old values of $x,y$ and $z$ with the new values found by using the equations from $($a$).$ In other words, you should have $x,$ say, appearing on both sides of your assignment statement. This will use the old values of $y$ and $z$ to calculate your new value of $x,$ etc. Then you calculate $y$ using the old values of $y$ and $z,$ but the new value of $x.$ When calculating $z$ you will use the old value of $z,$ but the new values of $x$ and $y.$

$($c$)$ Make sure you use the command format long before your loop, so that we get an approximation with a high level of accuracy. After your loop finishes, substitute the values found for $x,y$ and $z$ into the original LHS$\text{'}$s$,$ so we can compare them to the required RHS values.

How to write a matlab code for it$?$