5x - 3y + 2z = 20 .... pers 1

3x + 4y - 2z = 15 ..... pers 2

2x - 5y - 3z = 10 ..... pers 3

Eleminasi z pers l dan 2

5x - 3y + 2z = 20

3x + 4y - 2z = 15

----------------------- (+)

8x + y = 35 ...... pers 4

Eleminasi z pers l dan 3

3x + 4y - 2z = 15 |×3|

2x - 5y - 3z = 10 |×2|

9x + 12y - 6z = 45

4x - 10y - 6z = 20

-------------------------- (-)

5x + 22y = 25 ...... pers 5

Eleminasi pers 4 dan 5

8x + y = 35 |×22|

5x + 22y = 25 |×1|

176x + 22y = 770

5x + 22y = 25

--------------------- (-)

171x = 745

x = 745/171

Subtitusi x ke pers 4

8x + y = 35

8 (745/171) + y = 35

5960/171 + y = 35

y = 35 - 5960/171

y = 5985/171 - 5960/171

y = 25/171

Subtitusi x dan y ke pers 1

5x - 3y + 2z = 20

5 (745/171) - 3 (25/171) + 2z = 20

3725/171 - 75/171 + 2z = 20

3650/171 + 2z = 20

2z = 20 - 3650/171

2z = 3420/171 - 3650/171

2z = -230/171

z = -230/171 ÷ 2

z = -230/171 × 1/2

z = -230/342

z = -115/171

Maka, Himpunan penyelesaian sistem persamaan linear tiga variabel tersebut adalah;

HP = {(x, y, z)}

HP = {(745/171, 25/171, -115/171})

Pembuktian;

3x + 4y - 2z = 15

3 (745/171) + 4 (25/171) - 2 (-115/171) = 15

2235/171 + 100/171 + 230/171 = 15

2565/171 = 15

15 = 15 (terbukti)