Question

how do you find the coordinates of the point (x, y, z) on the plane z = 1 x + 1 y + 2 which is closest to the origin

Answer 1

the closest point is surely on a line from (0,0,0) meeting the plane at right angles.
but the normal to the given plane is (1,1,-1)
so the line must have direction (1,1,-1)
it's parametric equations would be:
x = k
y = k
z = -k

sub that in to the plane to find the value of k, sub back in the parametrics
and voila, there is your point.