I am working on solving a system of equations where I simplified to two equations and two unknowns. The two equations are nonlinear and complicated, and I am unsure how to solve it. Having done equations with easier, well-made algebraic solutions in college, I don't know how to work with more complicated equations. Here are the equations, please help me with this problem:
$$\left(A\times\sinh^{-1}\dfrac{\dfrac{L}{2}+\sinh\dfrac{a}{A}}{A}-a\right)L=27.625\left(L-\sinh\dfrac{a}{A}\right)+25.875A\\\dfrac{L}{A}=\sinh\dfrac{27.625+a}{A}-\sinh\dfrac{a}{A}$$ where $L$ is a given.
Thank you!
The second equation $$\dfrac{L}{A}=\sinh\left(\dfrac{t+a}{A}\right)-\sinh\left(\dfrac{a}{A}\right)\qquad \text{where} \qquad t=27.625$$ can rewrite as $$\dfrac{L}{A}=2\sinh\left(\dfrac{t}{A}\right)\cosh\left(\dfrac{t+2a}{A}\right)$$ from which we can eliminate $a$ to get two solutions $$a_{\pm}=-\frac{1}{2} \left(A \cosh ^{-1}\left(\frac{L }{2 A}\text{csch}\left(\frac{t}{A}\right)\right)\pm t\right)$$ For each case, replace this nice expression in the first equation which is now function of $A$ only. Plot them to locate more or less where is the solution and use Newton method with numerical derivatives (except if you enjoy nightmares).
Have fun !