Textbook----------

x + 3y + 5z = 4

x + 2y - 3z = 5

2x + 5y + 2z =8

This system has no solution.

The planes in the row picture don't meet at a point.

End of Textbook------------

Ok.

Three vectors fill a 3D space if all three at independent, ie they are not linear combinations of each other.

Three vectors fill a plane if 2 are independent and 1 is a linear combination of the other two.

Take equation one.

x + 3y + 5z = 4

which is the same as

x[1] + y[3] + z[5] = [4]

I think I may be retarded here...

[3] is a linear combination of [1] of the form 3[1]

Therefore x[1] + y[3] are dependent vectors. ***

have a feeling this might be wrong.

[5] is a linear combination of 2[1] and 3[1]

Therefore x[1] + y[3] + z[5] are all dependent vectors...

So how the heck do they make a plane? Wouldn't they just make a line?

If all three equations make lines, and the lines don't intersect, I understand. But the thing that's throwing me off is from the text: "The planes in the row picture don't meet at a point."

Is there something I'm not getting that is blatantly obvious?