How do you scetch z = x + y, what does it look like?

It's a plane passing through the origin. It contains all lines of the form c = x + y where c is a constant and has the vector (1, 1, -1) as a normal to it.

if it were a line in R2 you'd look for the x and y intercepts, which would be points, by letting x and y be 0 in turn and then solving for the other variable.

the equivalent in R3 is knowing that you have a plane which intersects the xy, the yz, and the xz planes in a line. when z = 0, you have the line y = -x in the xy plane. it goes through the origin with a slope of -1. let x = 0, and then z = y is a line in the yz plane, also through the origin. likewise, if y = 0, you have z = x, a line in the xz plane.

imagine you're standing on the xy plane, with the positive x axis to your right, the positive y axis in front of you, the positive z axis above you. you're holding a big sheet of glass with its bottom edge on the x axis, both sides of the origin. let's say positive x is east, positive y is north. now you rotate clockwise, keeping the bottom edge of the sheet of glass on the origin, until you're facing northeast. bottom edge of your glass is sitting on x + y = 0. now let the glass lean away from you until it's at a 45° angle with your floor (the xy plane), and let it expand without limit.

that's your plane.