Can three unknown variables be solved (a, b, c) from three equations?

[J0hn]

D - G = a(D-1) + b(D-1)² + c(D-1)³
E - H = a(E-1) + b(E-1)² + c(E-1)³
F - J = a(F-1) + b(F-1)² + c(F-1)³

Using the three equations above, how do I transpose to three separate equations for the unknown variables a, b, & c ?

a=?
b=?
c=?

a = (required equation can not contain the variables b or c but can contain all or any of the known variables)
b = (required equation can not contain the variables a or c but can contain all or any of the known variables)
c = (required equation can not contain the variables a or b but can contain all or any of the known variables)

Known Variables (I've given them values from measurements I've made)

D = 1.8926846
E = 2.5686994
F = 3.3756357

G = 1.892797681
H = 2.568917298
J = 3.376008599

D - G = -1.1308073E-04
E - H = -2.1789774E-04
F - J = -3.7289941E-04

D-1 = 0.8926846
E-1 = 1.5686994
F-1 = 2.3756357

Please note that all the known variables are from measurements made and from these measurements the unknown variables a, b, and c need to be calculated somehow from the three equations at the top of the page. Help !

 

[Deleted member 4993]

D - G = a(D-1) + b(D-1)² + c(D-1)³
E - H = a(E-1) + b(E-1)² + c(E-1)³
F - J = a(F-1) + b(F-1)² + c(F-1)³

Using the three equations above, how do I transpose to three separate equations for the unknown variables a, b, & c ?

a=?
b=?
c=?

a = (required equation can not contain the variables b or c but can contain all or any of the known variables)
b = (required equation can not contain the variables a or c but can contain all or any of the known variables)
c = (required equation can not contain the variables a or b but can contain all or any of the known variables)

Known Variables (I've given them values from measurements I've made)

D = 1.8926846
E = 2.5686994
F = 3.3756357

G = 1.892797681
H = 2.568917298
J = 3.376008599

D - G = -1.1308073E-04
E - H = -2.1789774E-04
F - J = -3.7289941E-04

D-1 = 0.8926846
E-1 = 1.5686994
F-1 = 2.3756357

Please note that all the known variables are from measurements made and from these measurements the unknown variables a, b, and c need to be calculated somehow from the three equations at the top of the page. Help !

You have three linear equations involving 3 unknowns (a, b & c). There are several "analytical solution method available. Are you allowed to use spreadsheet softwares like Excel?

Please show us what you have tried and exactly where you are stuck.​

​

Please follow the rules of posting in this forum, as enunciated at:​

​

https://www.freemathhelp.com/forum/threads/read-before-posting.109846/#post-486520​

​

Please share your work/thoughts about this assignment.​

 

[J0hn]

You have three linear equations involving 3 unknowns (a, b & c). There are several "analytical solution method available. Are you allowed to use spreadsheet softwares like Excel?

Please show us what you have tried and exactly where you are stuck.​

​

Please follow the rules of posting in this forum, as enunciated at:​

​

https://www.freemathhelp.com/forum/threads/read-before-posting.109846/#post-486520​

​

Please share your work/thoughts about this assignment.​

I always use excel but my level of mathematics is limited to calculating just two unknowns from three equations. I’m Completely stuck where to start with three Unknown’s.

 

[HallsofIvy]

D - G = a(D-1) + b(D-1)² + c(D-1)³
E - H = a(E-1) + b(E-1)² + c(E-1)³
F - J = a(F-1) + b(F-1)² + c(F-1)³

Using the three equations above, how do I transpose to three separate equations for the unknown variables a, b, & c ?

That is pretty much a straightforward system of linear equations. Solve for one a, b, or c at a time, eliminating the others by combining the equations to end with one equation in on unknown.

For example, I might choose to solve for a, eliminating first c, then b. If the first equation c is multiplied by \(\displaystyle (D- 1)^3\) and in the second equation by \(\displaystyle (E- 1)^3\). If we multiply the first by \(\displaystyle (E- 1)^3\) and the second by \(\displaystyle (D- 1)^3\) both will have \(\displaystyle (D- 1)^3(E- 1)^3c\) and when we subtract one equation from the other those will cancel.

\(\displaystyle (D - G)(E- 1)^3 = a(D-1)(E- 1)^3 + b(D-1)^2(E- 1)^3 + c(D-1)^3(E- 1)^3\)
\(\displaystyle (E - H)(D- 1)^3 = a(E-1)(D- 1)^3+ b(E-1)^2(D- 1)^3 + c(E-1)^3(E- 1)^3\)
Subtracting
(D- G)(E-1)^3- (E- H)(D-1)^3= a[(D-1)(E-1)^3- (E-1)(D-1)^3]+ b[(D-1)^2(E-1)^3- (E-1)^2(D-1)^3]

Do the same with the first and third (or second and third) equations to again eliminate c so that you have two equations in a and b. Then use the same idea to eliminate b, leaving one equation in a only. Solve that equation for a, then go back and use that value of a to solve for b, then use those values of a and b to solve for c.

a=?
b=?
c=?

a = (required equation can not contain the variables b or c but can contain all or any of the known variables)
b = (required equation can not contain the variables a or c but can contain all or any of the known variables)
c = (required equation can not contain the variables a or b but can contain all or any of the known variables)

Known Variables (I've given them values from measurements I've made)

D = 1.8926846
E = 2.5686994
F = 3.3756357

G = 1.892797681
H = 2.568917298
J = 3.376008599

D - G = -1.1308073E-04
E - H = -2.1789774E-04
F - J = -3.7289941E-04

D-1 = 0.8926846
E-1 = 1.5686994
F-1 = 2.3756357

Please note that all the known variables are from measurements made and from these measurements the unknown variables a, b, and c need to be calculated somehow from the three equations at the top of the page. Help !

 

[LCKurtz]

Like Halls says, this is just a linear system. I will save you some work and let Maple give you some answers. You can throw in some of your test numbers to check it:

The last equation is the one you want.

 

[J0hn]

Thank you HallsofIvy, I followed your advice and managed to transpose the equation to get rid of b and c and just leave variable a. I now have to follow the rest of your advise and do the same for b and c. Thank you very much for your help and pointing me in the right direction, much appreciated.

Thank you LCKurtz also.