Diketahui barisan sepuluh bilangan ganjil pertama, yaitu 1,3,5,7,9,11,13,15,17,19
Untuk menghitung penjumlahan 10 bilangan ganjil pertama maka gunakan deret aritmatika, yaitu
Persamaan Linear Tiga Variabel
Ax + By + Cz = D ………… (1)
Ex + Fy + Gz = H …………(2)
Px + Qy + Rz = S ………… (3)
Eliminasi persamaan (1) dan (2)
Ax + By + Cz = D * G AGx + BGy + CGz = DG
Ex + Fy + Gz = H * C ECx + FCy + GCz = HC -
(AG – EC)x + (BG – FC)y = DG – HC ............ (4)
Eliminasi persamaan (1) dan (3)
Ax + By + Cz = D *R ARx + BRy + CRz = DR
Px + Qy + Rz = S *C PCx + QCy + RCz = SC -
(AR – PC)x + (BR – QC)y = DR – SC ............... (5)
Eliminasi persamaan (4) dan (5)
(AG – EC)x + (BG – FC)y = DG – HC *(BR-QC) (AG – EC)(BR-QC)x + (BG – FC)(BR-QC)y = (DG – HC)(BR-QC)
(AR – PC)x + (BR – QC)y = DR – SC *(BG-FC) (AR – PC)(BG-FC)x + (BR – QC)(BG-FC)y = (DR – SC)(BG-FC) -
[(AG – EC)(BR-QC) – (AR – PC)(BG-FC)]x = (DG – HC)(BR-QC)
x = (DG – HC)(BR-QC) / [(AG – EC)(BR-QC) – (AR – PC)(BG-FC)]
Masukkan nilai x ke persamaan (4) atau (5)
misalnya ke persamaan (4)
(AG – EC)x + (BG – FC)y = DG – HC
(AG – EC) (DG – HC)(BR-QC) / [(AG – EC)(BR-QC) – (AR – PC)(BG-FC) + (BG – FC)y = DG – HC
y = (DG – HC)/ (BG – FC) - (AG – EC) (DG – HC)(BR-QC) / [(AG – EC)(BR-QC) – (AR – PC)(BG-FC) ] (BG – FC)
Masukkan nilai x dan y ke salah satu persamaan awal, persamaan (1), (2),atau (3)
misalnya ke persamaan (1)
Ax + By + Cz = D
A (DG – HC)(BR-QC) / [(AG – EC)(BR-QC) – (AR – PC)(BG-FC)] + B (DG – HC)/ (BG – FC) - (AG – EC) (DG – HC)(BR-QC) / [(AG - EC) (BR-QC) – (AR – PC)(BG-FC) ] (BG – FC) + Cz = D
z = D/C - [A (DG – HC)(BR-QC) / [(AG – EC)(BR-QC) – (AR – PC)(BG-FC)] + B ((DG – HC) / (BG – FC)) - (AG – EC) (DG – HC)(BR-QC) / [(AG - EC) (BR-QC) – (AR – PC)(BG-FC) ] (BG – FC)] /C
Jadi,
x = (DG – HC)(BR-QC) / [(AG – EC)(BR-QC) – (AR – PC)(BG-FC)]
y = (DG – HC)/ (BG – FC) - (AG – EC) (DG – HC)(BR-QC) / [(AG – EC)(BR-QC) – (AR – PC)(BG-FC) ] (BG – FC)
z = D/C - [A (DG – HC)(BR-QC) / [(AG – EC)(BR-QC) – (AR – PC)(BG-FC)] + B ((DG – HC) / (BG – FC)) - (AG – EC) (DG – HC)(BR-QC) / [(AG - EC) (BR-QC) – (AR – PC)(BG-FC) ] (BG – FC)] /C