On a piece of paper, sketch each of the following surfaces:

(i) $3 = x + y$

(ii) $3 = x + 3 y + z$

Use your graphs to fill in the following descriptions of cross-sections of the surfaces.

(a) For (i) ($3 = x + y$):
Cross sections with $x$ fixed give ? a horizontal line in a plane parallel to the yz-plane an empty set, or one or two vertical lines in a plane parallel to the yz-plane a vertical line in a plane parallel to the yz-plane two vertical lines in a plane parallel to the yz-plane a line in a plane parallel to the yz-plane an upward opening parabola in a plane parallel to the yz-plane a downward opening parabola in a plane parallel to the yz-plane a circle centered on the origin in a plane parallel to the yz-plane a point in a plane parallel to the yz-plane an empty set, or a circle centered on the origin in a plane parallel to the yz-plane
Cross sections with $y$ fixed give ? a horizontal line in a plane parallel to the xz-plane an empty set, or one or two vertical lines in a plane parallel to the xz-plane a vertical line in a plane parallel to the xz-plane two vertical lines in a plane parallel to the xz-plane a line in a plane parallel to the xz-plane an upward opening parabola in a plane parallel to the xz-plane a downward opening parabola in a plane parallel to the xz-plane a circle centered on the origin in a plane parallel the xz-plane a point in a plane parallel to the xz-plane an empty set, or a circle centered on the origin in a plane parallel to the xz-plane
Cross sections with $z$ fixed give ? a horizontal line in a plane parallel to the xy-plane two horizontal lines in a plane parallel to the xy-plane an empty set, or one or two horizontal lines in a plane parallel to the xy-plane a vertical line in a plane parallel to the xy-plane two vertical lines in a plane parallel to the xy-plane a line in a plane parallel to the xy-plane an upward opening parabola in a plane parallel to the xy-plane a downward opening parabola in a plane parallel to the xy-plane a circle centered on the origin in a plane parallel to the xy-plane a point in a plane parallel to the xy-plane an empty set, or a circle centered on the origin in a plane parallel to the xy-plane an empty set, or a plane parallel to the xy-plane

(b) For (ii) ($3 = x + 3 y + z$):
Cross sections with $x$ fixed give ? a horizontal line in a plane parallel to the yz-plane an empty set, or one or two vertical lines in a plane parallel to the yz-plane a vertical line in a plane parallel to the yz-plane two vertical lines in a plane parallel to the yz-plane a line in a plane parallel to the yz-plane an upward opening parabola in a plane parallel to the yz-plane a downward opening parabola in a plane parallel to the yz-plane a circle centered on the origin in a plane parallel to the yz-plane a point in a plane parallel to the yz-plane an empty set, or a circle centered on the origin in a plane parallel to the yz-plane
Cross sections with $y$ fixed give ? a horizontal line in a plane parallel to the xz-plane an empty set, or one or two vertical lines in a plane parallel to the xz-plane a vertical line in a plane parallel to the xz-plane two vertical lines in a plane parallel to the xz-plane a line in a plane parallel to the xz-plane an upward opening parabola in a plane parallel to the xz-plane a downward opening parabola in a plane parallel to the xz-plane a circle centered on the origin in a plane parallel the xz-plane a point in a plane parallel to the xz-plane an empty set, or a circle centered on the origin in a plane parallel to the xz-plane
Cross sections with $z$ fixed give ? a horizontal line in a plane parallel to the xy-plane two horizontal lines in a plane parallel to the xy-plane an empty set, or one or two horizontal lines in a plane parallel to the xy-plane a vertical line in a plane parallel to the xy-plane two vertical lines in a plane parallel to the xy-plane a line in a plane parallel to the xy-plane an upward opening parabola in a plane parallel to the xy-plane a downward opening parabola in a plane parallel to the xy-plane a circle centered on the origin in a plane parallel to the xy-plane a point in a plane parallel to the xy-plane an empty set, or a circle centered on the origin in a plane parallel to the xy-plane an empty set, or a plane parallel to the xy-plane