Menyelesaikan Sistem Persamaan Linear dengan Determinan

Kamis, 05 Juni 2014

Menyelesaikan Sistem Persamaan Linear dengan Determinan

Menyelesaikan Sistem Persamaan Linear dengan Determinan :

Sistem persamaan linear yang disusun dalam bentuk matriks juga dapat ditentukan himpunan penyelesaiannya dengan metode determinan. Misalnya, sistem persamaan linear untuk dua variabel dan tiga variabel adalah sebagai berikut.

a. ax + by = p
cx + dy = q

b. a₁x + b₁y + c₁z = d₁

a₂x + b₂y + c₂z = d₂

a₃x + b₃y + c₃z = d₃
Pada sistem persaman linear dua variabel, bentuk tersebut dapat diubah ke bentuk matriks berikut.

This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

, dengan A = This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

, X =

, dan B =

.

D =

= ad – bc (Determinan koefisien x dan y, dengan elemen-elemen matriks A)

Dx = This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

= pd – bq (Ganti kolom ke-1, dengan elemen-elemen matriks B)

Dy = This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

= aq – cp (Ganti kolom ke-2, dengan elemen-elemen matriks B)

Nilai x dan y dapat ditentukan dengan rumus berikut.
This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

Dengan cara yang sama dapat ditentukan D, D_x, D_y, dan D_z untuk sistem persamaan linear tiga variabel sebagai berikut.

Nilai x, y, dan z dapat ditentukan dengan cara berikut.

Contoh Soal 25 :

Tentukan penyelesaian sistem persamaan linear berikut dengan metode determinan.

a. 2x + y = 4
x – 2y = –3

b. x + y + z = 0
x + y – z = –2
x – y + z = 4

Penyelesaian :

a. Sistem persamaan linear di atas dapat disusun dalam bentuk matriks berikut.
This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

Kita tentukan nilai D, D_x, D_y .
D = This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

= – 4 – 1 = – 5
Dx = This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

= – 8 – (–3) = – 5
Dy = This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

= – 6 – 4 = – 10
Jadi, x = This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.

= 1 dan y = This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.