Question

tolongin kk" yg baikkk, mkasii•́⁠ ⁠ ⁠‿⁠ ⁠,⁠•̀​

Answer 1

Pertama, nyatakan SPLTV berikut ke dalam bentuk matriks.

[tex] \begin{cases} 4x+y+2z = 14 \\ 3x+2y+2z=16 \\ x+3y+3z = 17 \end{cases} [/tex]

Menjadi:

[tex] \displaystyle\begin{pmatrix} 4&1&2 \\ 3&2&2 \\ 1&3&3 \end{pmatrix} \begin{pmatrix} x\\y\\z \end{pmatrix} = \begin{pmatrix} 14\\16\\17 \end{pmatrix} [/tex]

Selanjutnya, tentukan [tex] D,D_x,D_y [/tex] dan [tex] D_z. [/tex] Dimulai dengan [tex] D: [/tex]

[tex] \begin{aligned} D& = \begin{array}{|ccc|} 4&1&2 \\ 3&2&2 \\ 1&3&3 \end{array} \begin{array}{cc} 4&1\\3&2\\1&3\end{array}\\ &= (4.2.3+1.2.1+2.3.3)-(2.2.1+4.2.3 \\ &\;\;\; \:+1.3.3)\\ &= (24+2+18)-( 4+24+9) \\ &= 44- 37 \\ &= 7\end{aligned} [/tex]

Kemudian [tex]D_x: [/tex]

[tex] \begin{aligned} D_x& = \begin{array}{|ccc|} 14&1&2 \\ 16&2&2 \\ 17&3&3 \end{array} \begin{array}{cc} 14&1\\16&2\\17&3\end{array}\\ &= (14.2.3+1.2.17+2.16.3)-(2.2.17 \\ &\;\;\; \:+14.2.3+1.16.3)\\ &= (84+34+96)-( 68+84+48) \\ &= 214- 200 \\ &= 14\end{aligned} [/tex]

Kemudian [tex] D_y: [/tex]

[tex] \begin{aligned} D_y& = \begin{array}{|ccc|} 4&14&2 \\ 3&16&2 \\ 1&17&3 \end{array} \begin{array}{cc} 4&14\\3&16\\1&17\end{array}\\ &= (4.16.3+14.2.1+2.3.17)-(2.16.1 \\ &\;\;\; \:+4.2.17+14.3.3)\\ &= (192+28+102)-( 32+136+126) \\ &= 322- 294 \\ &= 28\end{aligned} [/tex]

Terakhir tentukan [tex] D_z: [/tex]

[tex] \begin{aligned} D_z& = \begin{array}{|ccc|} 4&1&14 \\ 3&2&16 \\ 1&3&17 \end{array} \begin{array}{cc} 4&1\\3&2\\1&3\end{array}\\ &= (4.2.17+1.16.1+14.3.3)-(14.2.1\\ &\;\;\; \:+4.16.3+1.3.17)\\ &= (136+16+126)-(28+192+51) \\ &= 278-271 \\ &= 7\end{aligned} [/tex]

Maka, nilai [tex] x,y,z [/tex] nya adalah:

[tex] x= \frac{D_x}{D} = \frac{14}{7} = 2 [/tex]

[tex] y= \frac{D_y}{D}= \frac{28}{7} = 4 [/tex]

[tex] z= \frac{D_z}{D}= \frac77 = 1 [/tex]

Jadi, himpunan penyelesaian dari SPLTV tersebut adalah [tex] \text{HP} = \{(2,4,1) \} .[/tex]