<PRE><B>Question 335524</B>
3x+y+z=5
4x+5y-z=-8
10x+7y+z=2
Solve by Gauss-Jordan method


what we are going to do: use arithmetic operations to eliminate variables and put x, y, and z in a diagonal opposite their solutions



 3x +  y + z =  5
 4x + 5y - z = -8 (add to 1st equation)
10x + 7y + z =  2
 

 7x + 6y + 0z = -3
 4x + 5y -  z = -8
10x + 7y +  z =  2 (add to 2nd equation)


 7x +  6y + 0z = -3 (multiply both sides by -2)
14x + 12y + 0z = -6
10x +  7y +  z =  2


-14x - 12y + 0z =  6
 14x + 12y + 0z = -6 (add to 1st equation)
 10x +  7y +  z =  2


  0x +  0y + 0z =  0
 14x + 12y + 0z = -6 (2 variables here)
 10x +  7y +  z =  2 (3 variables here, we have a problem)


we still have 3 variables to solve for


we just lost one equation


we need 3 equations to solve for 3 variables


this has an infinite number of solutions


cases of having fewer equations than unknowns usually have an infinite number of solutions


these are referred to as underdetermined systems



this problem has an infinite number of solutions


if anyone finds an error in what I have done here please let me know


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REVISITING PROBLEM


  0x +  0y + 0z =  0
 14x + 12y + 0z = -6 (multiply both sides by 5)
 10x +  7y +  z =  2 (multiply both sides by -7)


  0x +   0y + 0z =   0
 70x +  60y + 0z = -30 (add to 3rd equation)
-70x -  49y - 7z = -14


  0x +   0y + 0z =   0
 70x +  60y + 0z = -30
  0x +  11y - 7z = -44


from 2nd equation:
60y = -70x - 30
 6y = - 7x -  3
  y = (-7/6)x - 3/6
  y = (-7/6)x - 1/2


check on 2nd equation:
70x + 60(-7/6)x - 60(1/2) = -30
70x + 10(-7)x - 30 = -30
70x - 70x - 30 = -30
0x - 30 = -30
-30 = -30


plug into 3rd equation:


0x + 11(-7/6)x - 11(1/2) - 7z = -44
0x - (77/6)x - (11/2) - 7z = -44
0x - (154/6)x - 11 - 14z = -88 (multiplied both sides by 2)
- (154/6)x - 14z = -77
-154x - 84z = -462 (multiplied both sides by 6)


-84z = 154x - 462
 84z = -154x + 462 (divided both sides by -1)
   z = (-154/84)x + (462/84)
   z = (-77/42)x + (231/42) (simplified both fractions by 2/2)
   z = (-11/6)x + (33/6) (simplified both fractions by 7/7)
   z = (-11/6)x + (11/2) (simplified 2nd fraction by 3/3)


now we got x = ? 
y = (-7/6)x - 1/2
(a line in x-y plane with slope -7/6 and y-intercept -1/2)
z = (-11/6)x + (11/2) 
(a line in x-z plane with slope -11/6 and z-intercept 11/2)


lets plug these into original equations:

  3x +  y + z =  5
  4x + 5y - z = -8
 10x + 7y + z =  2


  3x +   (-7/6)x - 1/2  + (-11/6)x + (11/2)     =  5
  4x + 5((-7/6)x - 1/2) -((-11/6)x + (11/2)) = -8
 10x + 7((-7/6)x - 1/2) + (-11/6)x + (11/2)  =  2


  3x - (7/6)x - (1/2) - (11/6)x + (11/2)  =  5
  4x - (35/6)x - (5/2) + (11/6)x - (11/2) = -8
 10x - (49/6)x - (7/2) - (11/6)x + (11/2) =  2


  3x - (18/6)x + (10/2) =  5
  4x - (24/6)x - (16/2) = -8
 10x - (60/6)x + (4/2)  =  2


  3x -  3x + 5 =  5
  4x -  4x - 8 = -8
 10x - 10x + 2 =  2


no solution for x
y = (-7/6)x - 1/2
(a line in x-y plane with slope -7/6 and y-intercept -1/2)
z = (-11/6)x + (11/2) 
(a line in x-z plane with slope -11/6 and z-intercept 11/2)


again, if you find an error in what I have done please let me know





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