Selesaikan soal berikut dengan menggunakan metode invers.

a. x = -34/11, y = 42/11; b. x = 19/8, y = 7/2

Pembahasan

Untuk menyelesaikan sistem persamaan linear dengan metode invers, kita ubah soal cerita ke dalam bentuk matriks.

a. Sistem persamaan: -2x + y = 10 dan 5x + 3y = -4
Dalam bentuk matriks: 
[[-2, 1], [5, 3]] [x, y] = [10, -4]

Untuk mencari invers dari matriks A = [[a, b], [c, d]] adalah A⁻¹ = (1/(ad-bc)) [[d, -b], [-c, a]].

Determinan D = (-2)(3) - (1)(5) = -6 - 5 = -11.

Invers A⁻¹ = (1/-11) [[3, -1], [-5, -2]] = [[-3/11, 1/11], [5/11, 2/11]].

[x, y] = A⁻¹ [10, -4]
[x, y] = [[-3/11, 1/11], [5/11, 2/11]] [10, -4]
[x, y] = [(-3/11)*10 + (1/11)*(-4), (5/11)*10 + (2/11)*(-4)]
[x, y] = [(-30 - 4)/11, (50 - 8)/11]
[x, y] = [-34/11, 42/11]

Jadi, x = -34/11 dan y = 42/11.

b. Sistem persamaan: 0x + 2y = 7 dan 4x - y = 6
Dalam bentuk matriks: 
[[0, 2], [4, -1]] [x, y] = [7, 6]

Determinan D = (0)(-1) - (2)(4) = 0 - 8 = -8.

Invers A⁻¹ = (1/-8) [[-1, -2], [-4, 0]] = [[1/8, 2/8], [4/8, 0]] = [[1/8, 1/4], [1/2, 0]].

[x, y] = A⁻¹ [7, 6]
[x, y] = [[1/8, 1/4], [1/2, 0]] [7, 6]
[x, y] = [(1/8)*7 + (1/4)*6, (1/2)*7 + 0*6]
[x, y] = [7/8 + 6/4, 7/2]
[x, y] = [7/8 + 12/8, 7/2]
[x, y] = [19/8, 7/2]

Jadi, x = 19/8 dan y = 7/2.

Jawaban:
a. x = -34/11, y = 42/11
b. x = 19/8, y = 7/2