software for symbolic math plotter I'm wondering if there is a plotting tool out there that will work in symbolic math? 
For example, if I plot $x+y=a$ and $y=x$ on a chart, I should be able to click on the point of intersection to get $\left(\large \frac{a}{2},\frac{a}{2}\right)$.
 A: Another soft for geometric drawings, which is maybe better than Geometer's Sketchpad, and (more important) free is Geogebra. In Geogebra you can plot lines given by certain equations (unfortunately, I don't think symbolic equations work), and you have a left panel with all the objects you have drawn and also their equations. There is a tool for intersection of two objects, and a new point is created. Every point can be found in the left panel with its coordinates. 
A program which does symbolic computation and symbolic plotting at a very advanced level is Maple. Unfortunately, this is not free.
To do symbolic plotting is impossible on computer, since all the data on the screen needs coordinates to look the way it is. The best compromise, which I think is available with Geogebra, is to introduce a parameter, which varies between some fixed values, and can be modified with a cursor. Maple has this function for sure; I know that I've used it a couple of times.
A: Are you familiar with Geometer's Sketchpad?
There's a student version for $29.95, and I believe a free trial version.  The resources are fabulous (tutorials, exchange forums, downloadable/accessible sketches created by others, lessons, etc.) and it is really quite user friendly. It may even be "usable" on-line (free). Not sure on that.  
I believe it also allows you to create java applets on web pages, but it's usually what I turn to for plotting, geometric constructions, determining points of intersection, area, etc. Your "sketches" can be saved, reloaded to edit, saved as image files, etc.
If you have an iPhone (and iPod touch) or an iPad, there's a program ($30?) called Spacetime that you can purchase to download and use on your device (making it essentially a graphing calculator). You're better off with the Wolfram App mentioned below if you're interested in access to mobile graphing and much more.
I mention this mainly because if you go to the Spacetime website, there's also a free desktop version of the program (platform for Mac and Windows), which allows plotting,solving equations, etc., but isn't as powerful as Sketchpad.  It probably doesn't do much more than Wolfram Alpha can do, but I believe you can save work with the desktop version and I don't know you can do that with Wolfram Alpha. For more details of the capabilities of Spacetime, see the Spacetime online manual.
By the way, for the iPhone (and iPod touch) and iPad, there's an app (almost free, $1.99 USD - one time payment) available for downloading mobile access to Wolfram Alpha to your i-Pad-Pod-phone. 
A: Another way is to use Maple 15, 16 or recently version 17. In this powerful software which is famous in symbolic computations, you can easily probe the point of intersection as well as other useful points in any certain plots. I could do that for you while $a=4$:


