Help! My System of Linear Equations is on a Rampage - It Won't Follow the Rules! How Do I Tame These Rebel Equations?
Hello, I'm grappling with an intriguing puzzle involving a system of two equations with two unknowns. Despite my attempts at the conventional substitution method, I find myself stuck with trivial or incorrect solutions for $y$. However, a twist in my approach emerges when I treat one of the constants as a variable instead of $x$, magically leading me to the correct solution for $y$. Can anyone unravel this mathematical enigma and shed light on the unexpected twist in my solution strategy?
I wonder if I am missing something here...
$\text{Equation 1}: 0 = - c_{1} + \frac{c_{2}}{c_{9} \cdot \left(\frac{1}{x} + \frac{1}{c_{9}}\right)\left(\frac{1}{\frac{1}{y} + \frac{1}{c_{8}}} + \frac{1}{\frac{1}{x} + \frac{1}{c_{9}}}\right)} - \frac{c_{4}}{c_{9} \left(y + \frac{1}{\frac{1}{x} + \frac{1}{c_{9}} + \frac{1}{c_{8}}}\right) \left(\frac{1}{x} + \frac{1}{c_{9}} + \frac{1}{c_{8}}\right)}$
$\text{Equation 2}: 0 = - \frac{c_{3}}{c_{8}} + \frac{- c_{3} + c_{4}}{y} + \frac{\left(c_{2} - c_{3}\right) \left(c_{9} + x\right)}{c_{9} x}$
Here is some python code and results to illustrate.
import numpy as np
import sympy as sym
c1,c2,c3,c4,c5,c6,c7,c8,c9,x,y,z = sym.symbols("c1,c2,c3,c4,c5,c6,c7,c8,c9,x,y,z")
equation1= -c1 + ((c2/((1/(1/c9 + 1/x))+(1/(1/y + 1/c8)))))*((1/c9)/(1/c9 + 1/x)) - ((c4/(y+(1/(1/x + 1/c8 + 1/c9))))*((1/c9)*(1/(1/x + 1/c8 + 1/c9))))
equation2 = ((-c3+c2)/((x*c9)/(x+c9))) + ((-c3+c4)/(y)) + ((-c3)/(c8))
solution_1_2 = sym.solve(equation1, x, dict=False)
print(sym.latex(solution_1_2))
solution_2_2 = sym.solve(equation2, x, dict=False)
print(sym.latex(solution_2_2))
solution_3_2 = sym.solve(solution_1_2[0]-solution_2_2[0], y, dict=False)
print(sym.latex(solution_3_2))
Solve equation 1 for x: $x= - \frac{c_{1} c_{8} c_{9} y}{c_{1} c_{8} c_{9} + c_{1} c_{8} y + c_{1} c_{9} y - c_{2} c_{8} - c_{2} y + c_{4} c_{8}} $
Solve Equation 2 for x: $x= \frac{c_{8} c_{9} y \left(c_{2} - c_{3}\right)}{- c_{2} c_{8} y + c_{3} c_{8} c_{9} + c_{3} c_{8} y + c_{3} c_{9} y - c_{4} c_{8} c_{9}} $
Substitute and Solve for y: $y=\left[ 0, \ \frac{c_{8} \left(- c_{2} + c_{4}\right)}{c_{2}}\right] $
import numpy as np
import sympy as sym
c1,c2,c3,c4,c5,c6,c7,c8,c9,x,y,z = sym.symbols("c1,c2,c3,c4,c5,c6,c7,c8,c9,x,y,z")
equation1= -c1 + ((c2/((1/(1/c9 + 1/x))+(1/(1/y + 1/c8)))))*((1/c9)/(1/c9 + 1/x)) - ((c4/(y+(1/(1/x + 1/c8 + 1/c9))))*((1/c9)*(1/(1/x + 1/c8 + 1/c9))))
equation2 = ((-c3+c2)/((x*c9)/(x+c9))) + ((-c3+c4)/(y)) + ((-c3)/(c8))
solution_1_2 = sym.solve(equation1, c9, dict=False)
print(sym.latex(solution_1_2))
solution_2_2 = sym.solve(equation2, c9, dict=False)
print(sym.latex(solution_2_2))
solution_3_2 = sym.solve(solution_1_2[0]-solution_2_2[0], y, dict=False)
print(sym.latex(solution_3_2))
Solve equation 1 for c9: $ c9=\frac{- c_{1} c_{8} x y + c_{2} c_{8} x + c_{2} x y - c_{4} c_{8} x}{c_{1} c_{8} x + c_{1} c_{8} y + c_{1} x y}$
Solve equation 2 for c9 $ c9=\frac{c_{8} x y \left(c_{2} - c_{3}\right)}{- c_{2} c_{8} y + c_{3} c_{8} x + c_{3} c_{8} y + c_{3} x y - c_{4} c_{8} x}$
Substitute and Solve for y: $ y=\left[\frac{c_{8} \left(- c_{2} + c_{4}\right)}{c_{2}}, \ \frac{c_{8} x \left(c_{3} - c_{4}\right)}{c_{1} c_{8} x + c_{2} c_{8} - c_{3} c_{8} - c_{3} x}\right] $