Solve 4 equations with 4 unknowns

[EssoExplJoe]

I have managed arrive a solution and programmed it in VBA for three quadratic equations and three unknown. However, I seem to be getting lost in weeds trying to write a general solution for a,b,c,d in the following:

Points: (Y₁,X₁), (Y₂,X₂), (Y₃,X₃), (Y₄,X₄)
Points: (10,2), (5,4), (4,11), (4,15)
Four Equations with 4 variables:
a) Y₁=a+bX₁+cX₁²+dX₁³
b) Y₂=a+bX₂+cX₂²+dX₂³
c) Y₃=a+bX₃+cX₃²+dX₃³
d) Y₄=a+bX₄+cX₄²+dX₄³

I am searching for the solution and was hoping someone had already done this.

 

[HallsofIvy]

(c) and (d) are possible if b+ c+ d= 0.

 

[Deleted member 4993]

I have managed arrive a solution and programmed it in VBA for three quadratic equations and three unknown. However, I seem to be getting lost in weeds trying to write a general solution for a,b,c,d in the following:

Points: (Y₁,X₁), (Y₂,X₂), (Y₃,X₃), (Y₄,X₄)
Points: (10,2), (5,4), (4,11), (4,15)
Four Equations with 4 variables:
a) Y₁=a+bX₁+cX₁²+dX₁³
b) Y₂=a+bX₂+cX₂²+dX₂³
c) Y₃=a+bX₃+cX₃²+dX₃³
d) Y₄=a+bX₄+cX₄²+dX₄³

I am searching for the solution and was hoping someone had already done this.

Use MS-excel and the "minverse" function.

 

[Deleted member 4993]

So are you saying that (using given data) the 4 equations are:

a) a + 2b + 4c + 8d = 10
b) a + 4b + 16c + 64d = 5
c) a + 11b + 121c + 1331d = 4
d) a + 15b + 225c + 3375d = 4

If so, how d'heck are c) and d) possible?

I get:

a =	18.78022
b =	-5.48834
c =	0.587413
d =	-0.01915

and back substitution gives me:

Y1	9.999992
Y2	4.999868
Y3	3.996803
Y4	3.991795

 

[Deleted member 4993]

(c) and (d) are possible if b+ c+ d= 0.

How did you get that condition?

 

[Deleted member 4993]

I have managed arrive a solution and programmed it in VBA for three quadratic equations and three unknown. However, I seem to be getting lost in weeds trying to write a general solution for a,b,c,d in the following:

Points: (Y₁,X₁), (Y₂,X₂), (Y₃,X₃), (Y₄,X₄)
Points: (10,2), (5,4), (4,11), (4,15)

There is an "algorithm" called Gauss-elimination & back substitution method. Easy to implement and stable. Look it up...

 

[EssoExplJoe]

Unknows

Solved by using Excel Minverse and MMult to arrive a a matrix solution!