Solve a system of equations with two unknowns

Hi everyone this should be a pretty easy question. I have two equations with two unknowns

1. -161.20 +2.7Ts -2.7Ta +3.91x10^-8 Ts^4 -5.67x10^-8 Ta^4 =0
2. -78.02 +2.7Ts -2.7Ta -3.57x10^-8 Ts^4 +1.13x10^-7 Ta^4 =0

where Ts and Ta are the unknowns I am trying to calculate. How do I work out the answer for this? Thanks for the help.

Answer 1

[Greg Heath]

With an obvious change of variables, the two equations can be combined to obtain

 y^4 =( 83.18 + 16.97x^4)/7.48

 and

 y = x +(117.72 -3.2007x^4)/270 

 leading to a 16th ? order polynomial equation in x.

Hope this helps.

Thank you for formally accepting my answer

Greg

[Greg Heath]

Given the two simultaneous equations

   y    = 0.436 +x -0.011854*x.^4   (1)
   y^4 = ( 11.12 + 2.2687*x.^4)      (2)

   and taking the fourth root of the latter yields 
   the following 4 equations which can be combined 
   with the first equation to eliminate y.

   y = (+/-)*  ( 11.12 + 2.2687*x.^4)^(1/4)   (3,4)
   y = (+/-)*i*( 11.12 + 2.2687*x.^4)^(1/4)   (5,6)

   Plotting equations 1,3 and 4 

    x  = -4:0.05:6;                                 
   y1 =  0.436 +x -0.011854*x.^4 ;         %(1)
   y2 =   ( 11.12 + 2.2687*x.^4).^(1/4) ;  %(2)
   y3 = -  ( 11.12 + 2.2687*x.^4).^(1/4);  %(3)

   close all, figure
   hold on 
   plot(x,y1,'k')
   plot(x,y2,'b')
   plot(x,y3,'r')

   Shows that although y2 is always greater than y1,
   y3 intersects y1 near  x = -3.25 and x = 5.75

   All other roots must be complex.

 Interesting.

 Greg

Answer 2

[Silibelo Kamwi]

Hi Greg, i guess you need to check the optimization toolbox, there is an example of a function dealing with this type of system of nonlinear equations, called fminunc and you might need to get the derivative of your system of equations.

cheers, IK