matrices, find intersection if any?

Question

x - y + 2z = 4
2x - 2y + 4z = 7
3x - 3y + 6z = 11

placing them in a matrix
1 -1 2 4
2 -2 4 7
3 -3 6 11

then proceeding with row-echelon form I got

1 -1 2 4
0 0 0 -1
0 0 0 -1

By analyzing the equations for the planes, I know that all the planes are parallel but are in different places.

What I'm having a problem with is that when I plug my equations into a program, I get:
1 -1 2 4
0 0 0 -1
0 0 0 0

What did I do wrong?

so what's with the program getting 0 0 0 0 on the 3rd role and me getting 0 0 0 -1 ?

Answer 1

you did nothing wrong...

it simply means the equations are inconsistent...

no values of x,y and z will satisfy all the equations simultaneously.

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Answer 2

You just didn't do the final step; your matrix is not quite in row-echelon form since the second and third rows both have a leading -1 in the fourth column.

If you now modify the third row by subtracting the second row from it, you'll get the same result as the program.