SOLUTION: 3x+y+z=5 4x+5y-z=-8 10x+7y+z=2 Solve by Gauss-Jordan method

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Question 335524: 3x+y+z=5
4x+5y-z=-8
10x+7y+z=2
Solve by Gauss-Jordan method
Answer by CharlesG2(834) About Me  (Show Source):
You can put this solution on YOUR website!
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