How the solution of a 3 variable equation is always a fixed plane?

Course Q&A Math for Machine Learning Linear Algebra for Machine Learning and Data Sc...
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visual link

In the above link for one of the equation - a+b+c =10 ( first equation in System 1)
There is a plane which contain the solution I have two questions regarding that.
1 Suppose i set c = 5, then a and b can form a plane which intersect z axis at 5
Also if I set c = 2 then it will be a plane intersecting z axis at 2 so how it a fixed “plane” is drawn in the visulization tool? And how can I draw this plane ?
please provide answer this concept is haunting me…

2 Why it will always be a plane ? Can’t it be a v shape plane in three dimension ?

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Any two points define a line.
Any three points define a plane. These are fundamental principles of geometry.

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