Question

Describe the surface in double-struck r3 represented by the equation x + y = 3. this is the set {(x, y, 3 − x − y)|x is in double-struck r, y is in double-struck r} which is a vertical plane that intersects the xy-plane in the line y = 3 − x, z = 0.

Answer 1

First, note that x+y=3 is a linear in x and y, and independent of z. So, this is just a line in the xy-plane, extended into and out of the plane. Since we have the line, we have all the points we need;  

the equation in R3 is a plane parallel to the z-axis and contains the line
y = 3 - x  
The plan sketched below.