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Membuat elementer 5 variabel dengan 5 persamaan

Membuat elementer 5 variabel dengan 5 persamaan

Sistem persamaan 5 variabel:

x1+x2+x3+x4+x5=102x1+x2+3x3+x4+2x5=15x1+3x2+2x3+4x4+x5=183x1+x2+x3+2x4+3x5=202x1+2x2+2x3+x4+x5=16\begin{aligned} x_1 + x_2 + x_3 + x_4 + x_5 &= 10 \\ 2x_1 + x_2 + 3x_3 + x_4 + 2x_5 &= 15 \\ x_1 + 3x_2 + 2x_3 + 4x_4 + x_5 &= 18 \\ 3x_1 + x_2 + x_3 + 2x_4 + 3x_5 &= 20 \\ 2x_1 + 2x_2 + 2x_3 + x_4 + x_5 &= 16 \end{aligned}
(1)

Matriks Augmentasi

[11111102131215132411831123202221116]\left[ \begin{array}{ccccc|c} 1 & 1 & 1 & 1 & 1 & 10 \\ 2 & 1 & 3 & 1 & 2 & 15 \\ 1 & 3 & 2 & 4 & 1 & 18 \\ 3 & 1 & 1 & 2 & 3 & 20 \\ 2 & 2 & 2 & 1 & 1 & 16 \end{array} \right]
(2)

  • Langkah 1 : Eliminasi x1x_1

    R2→R2−2R1[11111100−11−10−5132411831123202221116]R_2 \rightarrow R_2 - 2R_1 \left[ \begin{array}{ccccc|c} 1 & 1 & 1 & 1 & 1 & 10 \\ 0 & -1 & 1 & -1 & 0 & -5 \\ 1 & 3 & 2 & 4 & 1 & 18 \\ 3 & 1 & 1 & 2 & 3 & 20 \\ 2 & 2 & 2 & 1 & 1 & 16 \end{array} \right]
    (3)

    R3→R3−R1[11111100−11−10−502130831123202221116]R_3 \rightarrow R_3 - R_1 \left[ \begin{array}{ccccc|c} 1 & 1 & 1 & 1 & 1 & 10 \\ 0 & -1 & 1 & -1 & 0 & -5 \\ 0 & 2 & 1 & 3 & 0 & 8 \\ 3 & 1 & 1 & 2 & 3 & 20 \\ 2 & 2 & 2 & 1 & 1 & 16 \end{array} \right]
    (4)

    R4→R4−3R1[11111100−11−10−50213080−2−2−10−102221116]R_4 \rightarrow R_4 - 3R_1 \left[ \begin{array}{ccccc|c} 1 & 1 & 1 & 1 & 1 & 10 \\ 0 & -1 & 1 & -1 & 0 & -5 \\ 0 & 2 & 1 & 3 & 0 & 8 \\ 0 & -2 & -2 & -1 & 0 & -10 \\ 2 & 2 & 2 & 1 & 1 & 16 \end{array} \right]
    (5)

    R5→R5−2R1[11111100−11−10−50213080−2−2−10−10000−1−1−4]R_5 \rightarrow R_5 - 2R_1 \left[ \begin{array}{ccccc|c} 1 & 1 & 1 & 1 & 1 & 10 \\ 0 & -1 & 1 & -1 & 0 & -5 \\ 0 & 2 & 1 & 3 & 0 & 8 \\ 0 & -2 & -2 & -1 & 0 & -10 \\ 0 & 0 & 0 & -1 & -1 & -4 \end{array} \right]
    (6)

  • Langkah 2 : Jadikan pivot kedua = 1

    R2→−R2[111111001−11050213080−2−2−10−10000−1−1−4]R_2 \rightarrow -R_2 \left[ \begin{array}{ccccc|c} 1 & 1 & 1 & 1 & 1 & 10 \\ 0 & 1 & -1 & 1 & 0 & 5 \\ 0 & 2 & 1 & 3 & 0 & 8 \\ 0 & -2 & -2 & -1 & 0 & -10 \\ 0 & 0 & 0 & -1 & -1 & -4 \end{array} \right]
    (7)

  • Langkah 3 : Eliminasi x2x_2

    R3→R3−2R2[111111001−110500310−20−2−2−10−10000−1−1−4]R_3 \rightarrow R_3 - 2R_2 \left[ \begin{array}{ccccc|c} 1 & 1 & 1 & 1 & 1 & 10 \\ 0 & 1 & -1 & 1 & 0 & 5 \\ 0 & 0 & 3 & 1 & 0 & -2 \\ 0 & -2 & -2 & -1 & 0 & -10 \\ 0 & 0 & 0 & -1 & -1 & -4 \end{array} \right]
    (8)

    R4→R4+2R2[111111001−110500310−200−4100000−1−1−4]R_4 \rightarrow R_4 + 2R_2 \left[ \begin{array}{ccccc|c} 1 & 1 & 1 & 1 & 1 & 10 \\ 0 & 1 & -1 & 1 & 0 & 5 \\ 0 & 0 & 3 & 1 & 0 & -2 \\ 0 & 0 & -4 & 1 & 0 & 0 \\ 0 & 0 & 0 & -1 & -1 & -4 \end{array} \right]
    (9)

    R1→R1−R2[10201501−110500310−200−4100000−1−1−4]R_1 \rightarrow R_1 - R_2 \left[ \begin{array}{ccccc|c} 1 & 0 & 2 & 0 & 1 & 5 \\ 0 & 1 & -1 & 1 & 0 & 5 \\ 0 & 0 & 3 & 1 & 0 & -2 \\ 0 & 0 & -4 & 1 & 0 & 0 \\ 0 & 0 & 0 & -1 & -1 & -4 \end{array} \right]
    (10)

  • Langkah 3 : Eliminasi x3x_3

    R4→R4+43R3[10201501−110500310−2000730−83000−1−1−4]R_4 \rightarrow R_4 + \frac{4}{3}R_3 \left[ \begin{array}{ccccc|c} 1 & 0 & 2 & 0 & 1 & 5 \\ 0 & 1 & -1 & 1 & 0 & 5 \\ 0 & 0 & 3 & 1 & 0 & -2 \\ 0 & 0 & 0 & \frac{7}{3} & 0 & -\frac{8}{3} \\ 0 & 0 & 0 & -1 & -1 & -4 \end{array} \right]
    (11)

  • Langkah 4 : Eliminasi x4x_4

    R5→R5+37R4[10201501−110500310−2000730−830000−1−363]R_5 \rightarrow R_5 + \frac{3}{7}R_4 \left[ \begin{array}{ccccc|c} 1 & 0 & 2 & 0 & 1 & 5 \\ 0 & 1 & -1 & 1 & 0 & 5 \\ 0 & 0 & 3 & 1 & 0 & -2 \\ 0 & 0 & 0 & \frac{7}{3} & 0 & -\frac{8}{3} \\ 0 & 0 & 0 & 0 & -1 & -\frac{36}{3} \end{array} \right]
    (12)

  • Langkah 5 : Eliminasi x3x_3

    R3→13R3[10201501−1105001130−23000730−830000−1−363]R_3 \rightarrow \frac{1}{3}R_3 \left[ \begin{array}{ccccc|c} 1 & 0 & 2 & 0 & 1 & 5 \\ 0 & 1 & -1 & 1 & 0 & 5 \\ 0 & 0 & 1 & \frac{1}{3} & 0 & -\frac{2}{3} \\ 0 & 0 & 0 & \frac{7}{3} & 0 & -\frac{8}{3} \\ 0 & 0 & 0 & 0 & -1 & -\frac{36}{3} \end{array} \right]
    (13)

Subsutusi balik :

  • Dari baris 5 kita dapat −x5=−367-x_5 = -\frac{36}{7}, hilangkan - sehingga kita memperoleh x5=367x_5 = \frac{36}{7}

  • Dari baris 4 kita dapat 73x4=−83\frac{7}{3}x_4 = -\frac{8}{3}, kemudian sederhanakan sehingga diperoleh x4=−87x_4 = -\frac{8}{7}

  • Dibaris 3 kita dapat 3x3+x4=−23x_3 + x_4 = -2, kemudian kita subsitusikan:

    3x3+x4=−23x3+(−87)=−23x3=−67x3=−273x_3 + x_4 = -2\\ 3x_3 + (-\frac{8}{7}) = -2\\ 3x_3 = -\frac{6}{7}\\ x_3 = -\frac{2}{7}
    (14)

    sehingga diperoleh x3=−27x_3 = -\frac{2}{7}

  • Dibaris 2 dapat x2−x3+x4=5x_2 - x_3 + x_4 = 5, kemudian kita subsitusikan:

    x2−x3+x4=5x2−(−27)+(−87)=5x2+27−87=5x2−67=5x2=357+67x2=417x_2 - x_3 + x_4 = 5\\ x_2 - (-\frac{2}{7}) + (-\frac{8}{7}) = 5\\ x_2 + \frac{2}{7} - \frac{8}{7} = 5\\ x_2 - \frac{6}{7} = 5\\ x_2 = \frac{35}{7} + \frac{6}{7}\\ x_2 = \frac{41}{7}
    (15)

    sehingga diperoleh x2=417x_2 = \frac{41}{7}

  • Dibaris 1 kida mendapat x1​+2x3​+x5​=5x_1​ + 2x_3 ​+ x_5 ​= 5, kemudian kita subsitusikan:

    x1​+2x3​+x5​=5x1+2(−27)+367=5x1+(−47)+367=5x1−47+367=5x1+327=5x1=357−327x1=37x_1​ + 2x_3 ​+ x_5 ​= 5\\ x_1 + 2(-\frac{2}{7}) + \frac{36}{7} = 5\\ x_1 + (-\frac{4}{7}) + \frac{36}{7} = 5\\ x_1 - \frac{4}{7} + \frac{36}{7} = 5\\ x_1 + \frac{32}{7} = 5\\ x_1 = \frac{35}{7} - \frac{32}{7}\\ x_1 = \frac{3}{7}\\
    (16)

    sehingga diperoleh x1=37x_1 = \frac{3}{7}

Kesimpulannya: solusi: x1=37,x2=417,x3=−27,x4=−87,x5=367x_1 = \frac{3}{7}, x_2 = \frac{41}{7}, x_3 = -\frac{2}{7}, x_4 = -\frac{8}{7}, x_5 = \frac{36}{7}

x1​+x2​+x3​+x4​+x5=10​2x1​+x2​+x3​+x4​+x5​=11x1​+2x2​+x3​+x4​+x5=11​x1​+x2​+2x3​+x4​+x5​=13x1​+x2​+x3​+2x4​+x5=12​x_1 ​+ x2 ​+ x3 ​+ x4​ + x5 = 10\\ ​2x1 ​+ x2​ + x3 ​+ x4 ​+ x5 ​= 11\\ x1 ​+ 2x2 ​+ x3 ​+ x4 ​+ x5 = 11\\ ​x1 ​+ x2 ​+ 2x3 ​+ x4 ​+ x5​ = 13\\ x1 ​+ x2 ​+ x3 ​+ 2x4 ​+ x5 = 12​
(17)
Komputasi Aljabar Linear
Artimatika Matriks
Komputasi Aljabar Linear
Persamaan Linear