Tentukan himpunan penyelesaian dari sistem persamaan x+y+z=4 5x+4y+z=6 8x+10y+z=15

Mapel : Matematika
Bab : Sistem Persamaan Linear Tiga Variabel

Diketahui :
- x + y + z = 4
- 5x + 4y + z = 6
- 8x + 10y + z = 15

Ditanya :
- Nilai x = .....?
- Nilai y = .....?
- Nilai z = .....?

Penyelesaian :
- Nilai x :
x + y + z = 4
z = 4 - x - y

5x + 4y + z = 6
5x + 4y + 4 - x - y = 6
5x - x + 4y - y = 6 - 4
4x + 3y = 2

8x + 10y + z = 15
8x + 10y + 4 - x - y = 15
8x - x + 10y - y = 15 - 4
7x + 9y = 11

(4x + 3y = 2) × 3
(7x + 9y = 11) × 1

12x + 9y = 6
7x + 9y = 11
_________ —
5x = - 5
x = - 5/5
x = - 1

- Nilai y :
4x + 3y = 2
4(- 1) + 3y = 2
- 4 + 3y = 2
3y = 2 + 4
3y = 6
y = 6/3
y = 2

- Nilai z :
z = 4 - x - y
z = 4 - (- 1) - 2
z = 4 + 1 - 2
z = 5 - 2
z = 3

Maaf Kalau Salah

x+y+z =4. data 1
5x +4y +z =6 data 2
8x +10y +z =15. data 3
*dari 1 dan 2
→ x+y+z =4
5x + 4y +z = 6
dieliminasi hasilnya = -4x -3y = -2. data 4

*dari 2 dan 3
5x +4y +z =6
8x+10y +z = 15
dieliminasi hasilnya = -3x -6y = -9. data 5
*dari 4 dan 5
-4x-3y =-2
-3x-6y = -9
diliminasi (atas kali 2, bawah kali 1)
-8x-6y = -4
-3x-6y = -9
hasilnya = -5x = 5,,,,nilai x = -1
cari y dari data 4
→ -4x-3y = -2
→ -4.(-1) - 3y = -2
→ 4 -3y = -2,,,, -3y =-6,, nilai y = 2

cari z dari data 1
→ x + y + z = 4
→ (-1) + 2 + z = 4
→ 1 + z = 4
→ z = 3


jadi nilai x = -1 ,,, y= 2 dan z = 3