I'm only looking at the first octant. So I know 3x+2y+z=10 is a plane (3D triangle almost) in the first octant, and that the plan y=3x cuts. Thing is, I have the solution, and it says the volume is the right side of the 3D traingle (the right side of y=3x, or depends on how you're viewing it, above y=3x). It's kind of hard to explain, but I'm wondering why its not the left side (below y=3x).
The bounds for z are z = 0 to z = 10 - 3x - 2y. Projecting this solid onto the xy-plane (that is, z = 0) yields a region bounded by 3x + 2y = 10, y = 3x, and x = 0. ==> y = 3x to y = 5 - 3x/2 with x in [0, 10/9]. So, the volume is given by