Question

Решить систему линейных уравнений: а. методом Крамера б. методом Гаусса. сделать проверку. 3x-y+z=2 4x+y-z=12 x-y+8z=0

Answer 1

 Метод Гауса: Складывая 1и 2  ур-я: 7x=14   x=2    складывая 2и 3 : 5x+7z=12     5*2+7z=12     7z=2       z=2/7   2-y+8*2/7=0   y=30/7

Answer 2

Solution using the Cramer's Rule:

To solve the given system of linear equations using Cramer's Rule, we need to find the values of x, y, and z.

The given system of equations is: 3x - y + z = 2 4x + y - z = 12 x - y + 8z = 0

To find the value of x, we replace the first column of the coefficient matrix with the column of constants and find the determinant. Then we divide this determinant by the determinant of the coefficient matrix.

Let's calculate the determinant of the coefficient matrix:

| 3 -1 1 | | 4 1 -1 | | 1 -1 8 |

The determinant of the coefficient matrix is: det(A) = 3(1)(8) + (-1)(-1)(1) + 1(4)(-1) - 1(1)(1) - (-1)(4)(8) - 8(1)(-1) = 3(8) + 1 + (-4) - 1 + 32 + 8 = 24 + 1 - 4 - 1 + 32 + 8 = 60.

Now, let's calculate the determinant of the matrix obtained by replacing the first column with the column of constants:

| 2 -1 1 | | 12 1 -1 | | 0 -1 8 |

The determinant of this matrix is: det(Ax) = 2(1)(8) + (-1)(-1)(0) + 1(12)(-1) - (-1)(1)(0) - (-1)(12)(8) - 8(1)(-1) = 2(8) + 0 + (-12) - 0 + 96 + 8 = 16 - 12 + 96 + 8 = 108.

Now, let's calculate the value of x:

x = det(Ax) / det(A) = 108 / 60 = 1.8.

Similarly, we can find the values of y and z by replacing the second and third columns of the coefficient matrix with the column of constants and calculating the determinants.

Let's calculate the determinant of the matrix obtained by replacing the second column with the column of constants:

| 3 2 1 | | 4 0 -1 | | 1 1 8 |

The determinant of this matrix is: det(Ay) = 3(0)(8) + 2(1)(1) + 1(4)(8) - 1(1)(8) - 1(4)(0) - 8(1)(1) = 0 + 2 + 32 - 8 - 0 - 8 = 18.

Now, let's calculate the determinant of the matrix obtained by replacing the third column with the column of constants:

| 3 -1 2 |