In the Interactive Tool: Graphical Representation of Linear Systems with 3 variables , I noticed you can toggle between a line, plane, or all space for the Infinite selection for Number of solutions:
line with infinite solutions:
plane with infinite solutions:
My question is: why exactly do they differ based on their equations? I notice in the line case the solutions for each variable themselves are linear equations with the variable t; while in the plane case the solutions use 2 variables t and s. Does it have anything to do with the linear dependencies?
In the infinite, plane case:
4x - 2y + 2z = 8;
2x - 1y + 1z = -4;
-2x + y - z = 4;
1st equation = second + -1*third equation
In the infinite line case:
4x - 3y + 1z = -8;
2x + 1y + 3z = 2;
2x - 4y - 2z = -10;
the first equation is equal to the sum of the 2nd and third equations
Additionally, where are the t and s variables coming from?
Visualizing the meaning of all this requires understanding the geometric interpretation. If you take any linear equation in 3 variables like this:
ax + by + cz = d
That is the equation of a plane in 3-space. The vector (a, b, c) is the normal vector to the plane and d is the minimal distance from the origin to the plane.
So anytime you choose one such equation, you are defining a plane. Now think about how you can define 3 such equations and all the possibilities for how those three planes can intersect. Or not. It’s been a while since I watched the lectures for this course, but I’ve got to believe that Prof Serrano gives geometric examples to demonstrate the possibilities here.
The equations define everything, so of course it matters what the equations are. I must be missing your point. 
Hi, thanks for the reply. I guess what I’m wondering is is there a simple way of just looking at the equations in the screenshot and being able to tell that one would have infinite solutions via a line, and another infinite solutions via a plane (without graphing)?
Short answer is “no”. Well, unless you are some kind of intuitive mathematical genius who can solve equations just by looking at them.
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(I mean algebraically). With 2 dimensions we can simply see if 2 lines have the same slope and whether or not the y-intercept is the same (which would make them co-linear and parallel or not co-linear and parallel) to tell between equations with 0 solutions vs equations with infinite solutions; and similarly for lines of different slope we know there is 1 intersection: Is there a similar method for 3 dimensions? (Sorry if this has been covered, I’ll re watch the material)
To follow up on this in case anyone is curious, it is covered in videos later; basically when he introduces the concept of rank and compares it to the number of variables/systems of equations:
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