Solve system of equations

Question

Are there any good resources for solving systems of equations out there?

I tried to put this into wolfram alpha, but it doesn´t seem to work:

 solve
 [
   {
    x+a=s+p,
    y+b=q+t, 
    (x^2+y^2)^(1/2)+(a^2+b^2)^(1/2)=(s^2+t^2)^(1/2)+(p^2+q^2)^(1/2),
    x^2+y^2+a^2+b^2=(4/5)*(s^2+t^2+p^2+q^2)
   },
   {s, t, p, q}
 ]

Is there something wrong with the syntax?

Answer 1

You can simplify your life (and WA's too) taking into account the fact that $x,y,a,b$ are constants for the problem. So, defining some intermediate numbers, your equations write $$A=s+p$$ $$B=q+t$$ $$C=\sqrt{p^2+q^2}+\sqrt{s^2+t^2}$$ $$D=p^2+q^2+s^2+t^2$$ and the result is just a monster !

You can eliminate variables $s$ and $t$ using the first equations and you are left with two unpleasant equations in $p$ and $q$.

More than likely, they will be multiple solutions.

For illustration purposes, I used $A=3$, $B=7$,$C=\sqrt{13}+\sqrt{17}$, $D=30$. The solutions are $$\{s= 2,p= 1,q= 4,t= 3\}$$ $$\left\{s= \frac{64}{29},p= \frac{23}{29},q= \frac{102}{29},t= \frac{101}{29}\right\}$$ $$\left\{s= \frac{23}{29},p= \frac{64}{29},q= \frac{101}{29},t= \frac{102}{29}\right\}$$ $$\{s= 1,p= 2,q= 3,t= 4\}$$

Answer 2

Type

Solve [{ x+a==s+p && y+b==q+t && Sqrt[x^2+y^2]+Sqrt[a^2+b^2]==Sqrt[s^2+t^2]+Sqrt[p^2+q^2] && x^2+y^2+a^2+b^2=(4/5)*(s^2+t^2+p^2+q^2) }, {s, t, p, q}]

Then wolfram mathematica works correctly. Evaluation takes a few minutes... The answer is too long to display. It can print the answer and this takes also a few minutes.