# Difference between planes intersecting along a common line and coinciding

For any system of three equations in three variables, we know that if their planes intersect at a point then there is a unique solution, there are no solutions if any or all three planes are parallel to each other, and that there are infinitely many solutions if they intersect along a common line or all three coincide.

I just can't visualize the difference between the two cases when they intersect along a common line and when they coincide with each other. Isn't the number of solutions infinite in both cases? I have some problem with imagining the third dimension ($z$-axis) in my head, so please tell me in simple words: What is the difference between the infinite number of solutions when the three planes intersect along a common line and when they coincide? Is any of the three variables fixed in a line in any plane?

God, my head spins! What's the difference between the two kinds of infinite solutions here?

-

An example of two planes whose intersection is a line is $x=0$ (the $yz$-plane) and $y=0$ (the $xz$-plane). The intersection is the line $\{(0,0,z):z\in\mathbb R\}$ (the $z$-axis).

An example of two planes which coincide is $x=0$ and $x=0$. The intersection is the plane $x=0$ or $\{(0,y,z):y,z\in\mathbb R\}$ (the $yz$-plane).

In each case, there is an infinite number of points in the intersection. The difference is that one is a line and one is a plane, that is all.

-
Thanks Jasper.The example with coordinates helped –  Ivy Mike Dec 5 '12 at 4:45
A plane is two dimensional; i.e. it takes two numbers to describe the whereabouts of a point on the plane. (Compare the usual $xy$-coordinate system.)