It requires 3 numbers to designate point in 3 dimensions (that's why it is 3 dimensions!) If you write a single equation connecting those three numbers, then, given any two you could solve for the third. That's why a single equation, in 3 dimensions, represents a two dimensional object such as a plane.
One way you can "visualize" a plane is to use the "intercepts" like in 2 dimensional graphing of a line. Any point on the z-axis, for example, has both x and y equal to 0. In your example, x+ y+ z= 2, if x and y are both equal to 0, z= 2. The plane must contain the point (0, 0, 2). In fact, for that equation it is easy to see that the other two intercepts are (2, 0, 0) and (0, 2, 0). Those three points form a triangle and the portiona of the plane in the first "octant" (where the 3 variables are all positive) is the interior of that triangle. The plane itself "lies on" that triangle.
Defennnder's method, of looking at different of z, also works very nicely for non-linear functions in which the surfaces are not planes. It gives the "countour maps" just like geodesic maps of mountains that show altitude.
