simultaneous equations in three variables ,and any order

[sed_y]

i have this problem:

z= a*(x^c) + b *(y^d),
where x,y,z are the variables , a,c ,c,d constants.
i have 4 set of values for z, x, and y available i.e
z1=a*(x1^c) + b *(y1^d) ---1
z2=a*(x2^c) + b *(y2^d) ----- 2
z1=a*(x3^c) + b *(y3^d)----- 3
z1=a*(x4^c) + b *(y4^d) ------4

how do i solvethese to know a,b,c,d.
if i use polynomial fit, in matlab, it wld do for 2 variables, say z vs x , z vs y, and if i add those
2 equations, with coefficients divided by two, same thing when done for some other x,y,z gives diffrent coefficients.

similarly, if i solve 1 and 2 assuming c=1,d=1, c=2,d=2, etc, i get a,b which i get diffenet for =n 3 and 4, which shd be same.

so, anyone, with ideas? is there matrix method?or approximation method?

 

[Papabravo]

I'm going to go out on a limb here and offer the conjecture that there is no general solution to the problem. Furthermore if you find a particular solution to the problem I do not think it would be unique.

You did not specify the domain for variables x, y, z. Integers, Real Numbers, Complex Numbers, Other? Come to think of it you did not specify the domain of the constants either.

 

[sed_y]

hi

well, all a,b,c,d are cosntants.
actually, we have a sateliite system, where z is change in phase spred function, and x, y are change in temperatures of satellite subsystems, we want to predict how z changes withx, y from 4 sets of data. so, thts why i formulated the equation.
is the approcah correct?

 

[Papabravo]

I don't know if the approach is correct or not since I don't know what a "phase spred function" is or why it should be a function of temperatures raised to a certain power.

I know that a, b, c, and d are constants. You said that in your original post. If x and y are temperatures, then they belong to a subset of the set of real numbers. This implies that z also belongs to a subset of the set of real numbers. The question which I have asked and which you have not answered is "Are a, b, c, and d restricted to a subset of the integers or can they also be a subset of the real numbers?"

What I am pretty sure of is that the techniques for solving equations with parameters for exponents are pretty limited. AFAIK only linear equations can be solved, and yours are most assuredly highly nonlinear for c or d not equal to 1.

You seem to be trying to derive an empirical relationship from experimental data rather than solving a system of simultaneous equations. These are two entirely different problems.