SOLUTION: <pre> Solve the system: -x + 2y - 6z = 4 x + y + 2z = 3 2x + 3y + 2z = 5 What does the answer mean graphically? </pre>

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Question 556914: 
Solve the system:

-x + 2y - 6z = 4
 x +  y + 2z = 3
2x + 3y + 2z = 5

What does the answer mean graphically?

Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!
-x + 2y - 6z = 4
 x +  y + 2z = 3
2x + 3y + 2z = 5

Eliminate x from the first two equations simply by adding them
like term by like term

-x + 2y - 6z = 4
 x +  y + 2z = 3
----------------
     3y - 4z = 7

Eliminate x from the first and third equations simply by multiplying
the first equation by 2

-2x + 4y - 12z = 8

and add it to the original third equation term by like term

 2x + 3y +  2z = 5
-2x + 4y - 12z = 8
------------------
      7y - 10z = 13

Now we put the two equations without x together

      3y -  4z =  7
      7y - 10z = 13

Eliminate z from those two equations by multiplying the first
one by -5 and the second one by 2, and adding term by term:

     -15y + 20z = -35
      14y - 20z =  26
      ---------------
       -y       =  -9
              y = 9

Substitute in

       7y - 10z =  13
     7(9) - 10z =  13
       63 - 10z =  13
           -10z = -50   
              z = 5

Substitute 9 for y and 5 for z in one of the
original equations, say the second one:

   x + y + 2z = 3
 x + 9 + 2(5) = 3  
   x + 9 + 10 = 3
       x + 19 = 3 
            x = -16

(x,y,z) = (-16,9,5)

Graphically that means that three planes intersect in the
3-dimensional point (-16,9,5)

Edwin
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