Sistem Persamaan Linier Tiga Variabel

Algoritma

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