4x-y+2z=5
2y+z=4
4x+y+3z=10
4x-1y+2z = 5
0x+2y+1z = 4
4x+1y+3z = 10
Erase all the letters
4 -1 +2 = 5
0 +2 +1 = 4
4 +1 +3 = 10
Erase all the + signs
4 -1 2 = 5
0 2 1 = 4
4 1 3 = 10
Replace the equal signs with a |
⎾4 -1 2 | 5⏋
⎢0 2 1 | 4⎢
⎿4 1 3 | 10⏌
We want to end up with a matrix that looks like this:
⎾1 0 0 | ⎽⏋
⎢0 1 0 | ⎽⎢
⎿0 0 1 | ⎽⏌
-------------------------
First we get all the 0's
⎾4 -1 2 | 5⏋
⎢0 2 1 | 4⎢
⎿4 1 3 | 10⏌
To get a 0 where the 4 is on the lower left hand corner,
multiply the top row by -1
⎾-4 1 -2 | -5⏋
⎢ 0 2 1 | 4⎢
⎿ 4 1 3 | 10⏌
Add the first row to the bottom row:
⎾-4 1 -2 | -5⏋
⎢ 0 2 1 | 4⎢
⎿ 0 2 1 | 5⏌
To get a 0 where the 1 is in the top row, multiply
the top row by -2
⎾ 8 -2 4 | 10⏋
⎢ 0 2 1 | 4⎢
⎿ 0 2 1 | 5⏌
Add the middle row to the top row:
⎾ 8 0 5 | 14⏋
⎢ 0 2 1 | 4⎢
⎿ 0 2 1 | 5⏌
To get a 0 where the 2 on the bottom row is, multiply
the bottom row by -1
⎾ 8 0 5 | 14⏋
⎢ 0 2 1 | 4⎢
⎿ 0 -2 -1 | -5⏌
Add the middle row to the bottom row
⎾ 8 0 5 | 14⏋
⎢ 0 2 1 | 4⎢
⎿ 0 0 0 | -1⏌
If you replace the letters now
8x + 0y + 6z = 14
0x + 2y + 1z = 4
0x + 0y + 0z = 1
The bottom equation has no solution because the left
side is 0 and the right side is 1, so there can be no
solution.
Edwin