# 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?

## 2 Answers

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.

A line is one dimensional, meaning it only takes one number to describe the position of a point on the line.

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.)

Three planes intersecting along a line corresponds to a one-dimensional solution. Only one variable (or parameter) is required to describe the complete solution. If all three planes coincide, it takes two variables (parameters) to describe the full solution.

In both cases, the number of solutions is infinite, but in a certain sense the solution set is ''larger'' in the second case.