8 equations 8 unknowns Simplify into 3 equations with only 3

8 equations, 8 unknowns

Simplify into (3) equations, with only 3 unknown variables:

F2 + F8 = F3

F3 + F4 = F5

F6 + F7 = F5

F8 + F9 = F7

0.95*F7*H7 = F9

2*F4 = F9 + F6*H7

F2*O2 = 2*F4 + F6*O6

F3*O3 – 2F4 = F5*O6

Solution

We assume that the 8 unknowns are F₂-F₉ and that H₇, O₂, O₃ and O₆ are scalars(coefficients). We will eliminate F₉,F₈,F₇,F₆ and F₅ as under:

       6.   2*F₄ = F₉ + F₆*H₇

       7. F₂*O₂ = 2*F₄ + F₆*O₆

       8. F₃*O₃ – 2F₄ = F₅*O₆

From the 4^th equation, we have F₉ =   F₇–F₈ .We will substitute this value of F₉ in the 5^th and 6^th equations to get

       9. 0.95*F₇*H₇ = F₇–F₈ or, F₈ = ( 1-0.95*H₇) F₇

      10. 2*F₄ = F₇–F₈ + F₆*H₇

From the 1^st equation, we have F₈ = F₃ –F₂. On substituting this value of F₈ in the 9^th and 10^th equations, we get

    11. F₃ –F₂ = ( 1-0.95*H₇) F₇

     12. 2*F₄ = F₇ - F₃ +F₂ + F₆*H₇

From the 3^rd equation, we get F₇ = F₅ –F₆. On substituting this value of F₇ in the 11^th and 12^th equations, we get

    13. F₃ –F₂ = ( 1-0.95*H₇)( F₅ –F₆)

    14. 2*F₄ = F₅ –F₆ - F₃ +F₂ + F₆*H₇ = F₂ –F₃ +(H₇ -1)F₆

From the 7^th equation, we get F₂*O₂ - 2*F₄ = F₆*O₆ or, F₆ = (1/ O₆ )(F₂*O₂ - 2*F₄). On substituting this value of F₆ in the 13^th and 14^th equations, we get

15. F₃–F₂=(1-0.95*H₇)[F₅ -(1/O₆)(F₂*O₂ -2*F₄)] or, F₃–[1-( 1-0.95*H₇)/O₆] F₂=(1-0.95*H₇)[F₅ +(2/O₆)F₄]

    16. 2*F₄ = F₂ –F₃ + [(H₇ -1)/ O₆ )(F₂*O₂ - 2*F₄) or, [1+ {(H₇ -1)*(O₂/O₆)]F₂ –F₃ = 2* [1- (H₇ -1)/ O₆]F₄

From the 2^nd equation, we get   F₅ = F₃ + F₄ .On substituting this value of F₅ in the 8^th and 15^th equations, we get

  17. F₃*O₃ – 2F₄ = O₆(F₃ + F₄) or, (O₃ –O₆)F₃ = (2+O₆)F₄

18. F₃ –[1-( 1-0.95*H₇)/O₆] F₂= (1-0.95*H₇)[( F₃ + F₄) +(2/O₆)F₄] or, 0.95*H₇*F₃–[1-( 1-0.95*H₇)/O₆] F₂   = (1-0.95*H₇)(1=2/O₆)F₄.

The required equations are:

16. [1+ {(H₇ -1)*(O₂/O₆)]F₂ –F₃ = 2* [1- (H₇ -1)/ O₆]F₄

17. (O₃ –O₆)F₃ = (2+O₆)F₄

18. 0.95*H₇*F₃–[1-( 1-0.95*H₇)/O₆] F₂   = (1-0.95*H₇)(1=2/O₆)F₄.