You have an issue withbecause the graphics is empty because not one single damn triple of {x,y,z} which gets tested fulfills the linear equation exactly. You remove the issue by displaying the half space typing
If you really want to plot a plane, then you use ParametricPlot3D[], if one knows two base vectors (i.e. linear independent vectors) in the plane, then one types
Please note, the vectors {0,1,3} and {1, 1/2, -1/3} are taken only for demonstration. You should compute the base vectors of the planes given to you on your own.

I have not yet found a way to get WolframAlpha to plot two planes, or any other set of 3d functions, in the same graph, all attempts end up with it plotting each function in a different graph.

I did find one possible way to get around this, first solve each equation for z and then plot3d the Max of the equations. If the intersection is obvious enough then this might provide the desired information.

As an example, this works for your last two equations http://www.wolframalpha.com/input/?i=Plot3D%5BMax%5B-y%2F2+%2B+2%2C+-x%2F3%2B2%5D%5D and should work for any number of equations as long as each include z. If some do not include z then perhaps you can interpret those equations appropriately and still use this method.

Still, if you do
in W|A you get two 3D plots