Let $S$ denote the yellow region, shown below. [asy]
unitsize(0.5 cm);

int x, y;

fill((0,8)--(0,6)--(2,2)--(4,1)--(9 ,0)--(10,0)--(10,8)--cycle, yellow);

for (x = 1; x <= 10; ++x) {
draw((x,0)--(x,8),gray(0.7));
}

for (y = 1; y <= 8; ++y) {
draw((0,y)--(10,y),gray(0.7));
}

draw((0,0)--(10,0));
draw((0,0)--(0,8));
draw((0,8)--(0,6)--(2,2)--(4,1)--(9 ,0)--(10,0)--(10,8)--cycle);

label("$x$", (10,0), E);
label("$y$", (0,8), N);
label("$0$", (0,0), S);
label("$10$", (10,0), S);
label("$0$", (0,0), W);
label("$8$", (0,8), W);

dot((0,6));
dot((2,2));
dot((4,1));
dot((9,0));
[/asy]

(a) For all points $(x, y)$ in $S,$ what is the minimum value of $x + y$?

(b) For all points $(x, y)$ in $S,$ what is the minimum value of $x + 3y$?