Tentukanlah penyelesaian sistem persamaan linear berikut

Penyelesaiannya adalah x = -2 dan y = 1.

Pembahasan

Untuk menyelesaikan sistem persamaan linear:
1) x - 2y + 4 = 0  => x - 2y = -4
2) 2x + y + 3 = 0  => 2x + y = -3

Dalam bentuk matriks Ax = B:
[ 1  -2 ] [ x ] = [ -4 ]
[ 2   1 ] [ y ] = [ -3 ]

A = [[1, -2], [2, 1]], x = [[x], [y]], B = [[-4], [-3]]

Metode Invers Matriks:
1. Cari determinan matriks A (det(A)).
   det(A) = (1 * 1) - (-2 * 2) = 1 - (-4) = 1 + 4 = 5

2. Cari invers matriks A (A^-1).
   A^-1 = (1/det(A)) * [[1, 2], [-2, 1]] = (1/5) * [[1, 2], [-2, 1]] = [[1/5, 2/5], [-2/5, 1/5]]

3. Cari solusi x = A^-1 * B.
   [ x ] = [[1/5, 2/5], [-2/5, 1/5]] * [[-4], [-3]]
   [ y ]
   x = (1/5)*(-4) + (2/5)*(-3) = -4/5 - 6/5 = -10/5 = -2
   y = (-2/5)*(-4) + (1/5)*(-3) = 8/5 - 3/5 = 5/5 = 1

Aturan Cramer:
1. Hitung det(A) = 5 (sudah dihitung di atas).

2. Hitung determinan Dx (ganti kolom x dengan B).
   Dx = det([[-4, -2], [-3, 1]]) = (-4 * 1) - (-2 * -3) = -4 - 6 = -10

3. Hitung determinan Dy (ganti kolom y dengan B).
   Dy = det([[1, -4], [2, -3]]) = (1 * -3) - (-4 * 2) = -3 - (-8) = -3 + 8 = 5

4. Cari solusi.
   x = Dx / det(A) = -10 / 5 = -2
   y = Dy / det(A) = 5 / 5 = 1

Jadi, penyelesaiannya adalah x = -2 dan y = 1.