I have these equations I was working on that although seemed simple, turned out to be deceptively hard.
I need to solve for r, x, and y in terms of s1, s2, s3, x1, x2, x3, y1, y2, y3.
I tried solving by hand and was able to get it down to just one equation with one unknown but the equation was so complicated I could not figure out how to solve it. Perhaps my approach was not correct, could anyone give me some pointers?
Thanks!
We can easily remove the r from the first equation and simplify it to two equations.
We can take the first equation above and solve for Y:
s1^4*x^2-s1^4*2x*x1+s1^4*x1^2+s1^4*y^2-s1^4*2y*y1+s1^4*y1^2 = s2^4*x^2-s2^4*2x*x2+s2^4*x2^2+s2^4*y^2-s2^4*2y*y2+s2^4*y2^2
s1^4*y^2-s1^4*2y*y1-s2^4*y^2+s2^4*2y*y2 = s2^4*x^2-s2^4*2x*x2+s2^4*x2^2+s2^4*y2^2-s1^4*x^2+s1^4*2x*x1-s1^4*x1^2+s1^4*2y*y1-s1^4*y1^2
(s1^4-s2^4)*y^2+(2*s2^4*y1-2*s1^4*y2)*y = s2^4*(x^2-2x*x2+x2^2+y2^2) - s1^4(x^2-2x*x1+x1^2+y1^2)
(s1^4-s2^4)*y^2+(2*s2^4*y1-2*s1^4*y2)*y + (s1^4(x^2-2x*x1+x1^2+y1^2) - s2^4*(x^2-2x*x2+x2^2+y2^2)) = 0
y = (2*s1^4*y2-2*s2^4*y1 \pm{\sqrt{(2*s2^4*y1-2*s1^4*y2)^2-4*(s1^4-s2^4)*s1^4(x^2-2x*x1+x1^2+y1^2) - s2^4*(x^2-2x*x2+x2^2+y2^2)}})/(2s1^4-2s2^4)
We can solve for Y of the second equation the same way giving us:
y = (2*s1^4*y3-2*s3^4*y1 \pm{\sqrt{(2*s3^4*y1-2*s1^4*y3)^2-4*(s1^4-s3^4)*s1^4(x^2-2x*x1+x1^2+y1^2) - s3^4*(x^2-2x*x3+x3^2+y3^2)}})/(2s1^4-2s3^4)
By combining these two equations we get:
(2*s1^4*y2-2*s2^4*y1 \pm{\sqrt{(2*s2^4*y1-2*s1^4*y2)^2-4*(s1^4-s2^4)*s1^4(x^2-2x*x1+x1^2+y1^2) - s2^4*(x^2-2x*x2+x2^2+y2^2)}})/(2s1^4-2s2^4) = (2*s1^4*y3-2*s3^4*y1 \pm{\sqrt{(2*s3^4*y1-2*s1^4*y3)^2-4*(s1^4-s3^4)*s1^4(x^2-2x*x1+x1^2+y1^2) - s3^4*(x^2-2x*x3+x3^2+y3^2)}})/(2s1^4-2s3^4)
This is one equation with only one unknown (x). However solving this for x eludes me.
edit:sorry the equations looks so bad, I tried putting it in the pretty math viewer you have but the equations were too big to fit.
edit: here is as far as I can get in solving that equation:
I can make this more readable by replacing some sections with shorthand variables. For instance:
And we have:
Since the c's are the only ones that contains the unknown, we can seperate that.
so basically I don't know how to combine those two sqrt's to solve for the variable in the c's.
Well i got a little bit farther, I got a very large equation and put that in a ti-89 emulator set to run super fast, and it's been working for nearly 24 hours now and I don't have confidance it will give me an answer. I am thinking there must be somethign wrong in my approach of this problem, but I cannot think of a way to solve it.
And I do thank VonNemo for his reply, but I do need this solved in terms of variables excluding x, y, and r.