# Reduce system of nonlinear equations to one nonlinear equation

I have a system of nonlinear equations:

L1^2 == (-Ay x2 + By x2 + Ax x3 - Bx x3 + Ay Bx x4 - Ax By x4)^2 / (x1^2 (By x2 - Bx x3 + x3 Rx - By x4 Rx - x2 Ry + Bx x4 Ry)^2) +
(-Ay Bx + Ax By + Ay Rx - By Rx - Ax Ry + Bx Ry)^2 / (By x2 - Bx x3 + x3 Rx - By x4 Rx - x2 Ry + Bx x4 Ry)^2

L2^2 == (-Cy x2 + Dy x2 + Cx x3 - Dx x3 + Cy Dx x4 - Cx Dy x4)^2 / (x1^2 (Dy x2 - Dx x3 + x3 Rx - Dy x4 Rx - x2 Ry + Dx x4 Ry)^2)
+ (-Cy Dx + Cx Dy + Cy Rx - Dy Rx - Cx Ry + Dx Ry)^2 / (Dy x2 - Dx x3 + x3 Rx - Dy x4 Rx - x2 Ry + Dx x4 Ry)^2

L3^2 == (-Ey x2 + Fy x2 + Ex x3 - Fx x3 + Ey Fx x4 - Ex Fy x4)^2 / (x1^2 (Fy x2 - Fx x3 + x3 Rx - Fy x4 Rx - x2 Ry + Fx x4 Ry)^2) +
(-Ey Fx + Ex Fy + Ey Rx - Fy Rx - Ex Ry + Fx Ry)^2 / (Fy x2 - Fx x3 + x3 Rx - Fy x4 Rx - x2 Ry + Fx x4 Ry)^2

L4^2 == (-Gy x2 + Hy x2 + Gx x3 - Hx x3 + Gy Hx x4 - Gx Hy x4)^2 / (x1^2 (Hy x2 - Hx x3 + x3 Rx - Hy x4 Rx - x2 Ry + Hx x4 Ry)^2) +
(-Gy Hx + Gx Hy + Gy Rx - Hy Rx - Gx Ry + Hx Ry)^2 / (Hy x2 - Hx x3 + x3 Rx - Hy x4 Rx - x2 Ry + Hx x4 Ry)^2


I need to solve it for x1, x2, x3, x4, other parameters will be given. I will solve this system in production in my JavaScript code, so the solution has to be found as fast as it possible. So how to simplify this equation in Wolfram Mathematica for fastest root finding?

• It is not all that clear what is wanted here. The system will have many solutions. Is there some criterion that might be used to give a guess for a root-finder? – Daniel Lichtblau Sep 25 at 15:31
• Could you provide sample values for the parameters, so that readers can consider possibilities? – bbgodfrey Sep 25 at 19:38