Question 1002469
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3x + 2y - 8z =  29
9x -  y + 2z =  23
-x - 2y + 8z = -11
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        When an assignment is to solve 3x3-system of equations, the goal is usually to teach a student to something useful.

        For example, to teach to look at equations and to search for something that makes the solution easy and educative.

        As usual (if the creators of the assignment are professionals), the goal is not simply manipulate with numbers,

        but to find some useful pattern.


        In this system, the first and the third equations have similar parts "2y - 8z" and "-2y + 8z",  

        so if we add these equations, we eliminate 'y' and 'z' simultaneously and easy will find 'z'.


        It is the key to start.



<pre>
So, your original system of equation is

    3x + 2y - 8z =  29    (1)
    9x -  y + 2z =  23    (2)
    -x - 2y + 8z = -11    (3)


Add equations (1) and (3).  The terms with 'y' and 'z' will cancel each other, and you will get then

    3x - x = 29 - 11,  --->  2x = 18,  --->  x = 18/2 = 9.


Now substitute this value x = 9 into the first and second equations

    3*9 + 2y - 8z = 29     (1')
    9*9 -  y + 2z = 23     (2')


Simplify

          2y - 8z =   2    (1'')
          -y + 2z = -58    (2'')


Now we are on the straight finish line: we only need to solve one 2x2-syatem of equations (1''), (2'').

Solve it by the Elimination method. For it, multiply equation (2'') by 2 (both sides) and add to equation (1'').
You will get

             - 8z + 4z = 2 + 2*(-58),  --->  -4z = -114,  z = (-114)/(-4) = 28.5.


Now from equation (2'')

           y = 58 + 2*28.5 = 115.


<U>ANSWER</U>.  The solution to the given system is x= 9,  y = 115,  z = 28.5.
</pre>

Solved.


I hope that you will learn something useful from my solution.