Question

Решить методом крамера x + 2y - z = 9 2x - y + 3z = 13 3x + 2y - 5z = -1

Answer 1

Решение на фотографии

Answer 2

To solve the system of linear equations using Cramer's method, we need to find the values of variables x, y, and z. Cramer's method involves finding determinants of matrices. Let's proceed step by step:

Step 1: Write the system of equations in matrix form:

Copy code
| 1  2 -1 |   | x |   |  9 |
| 2 -1  3 | * | y | = | 13 |
| 3  2 -5 |   | z |   | -1 |

Step 2: Find the determinant of the coefficient matrix (D):

scssCopy code
D = | 1  2 -1 |
    | 2 -1  3 |
    | 3  2 -5 |

D = 1 * ((-1) * (-5) - 3 * 2) - 2 * ((2 * (-5) - 3 * 3) + (-1) * 3) + (-1) * (2 * 2 - (-1) * 3)
D = 1 * (5 - 6) - 2 * (-10 - 9) - (-1 * 7)
D = -1 - (-38) - 7
D = -1 + 38 - 7
D = 30

Step 3: Find the determinant when we replace the x-column with the constant column (Dx):

scssCopy code
Dx = | 9  2 -1 |
     | 13 -1  3 |
     | -1  2 -5 |

Dx = 9 * ((-1) * (-5) - 3 * 2) - 2 * ((13 * (-5) - 3 * (-1)) + (-1) * 3) + (-1) * (13 * 2 - (-1) * (-1))
Dx = 9 * (5 - 6) - 2 * ((-65 + 3) - 3) + (-1) * (26 - 1)
Dx = 9 * (-1) - 2 * (-65) + 1
Dx = -9 + 130 + 1
Dx = 122

Step 4: Find the determinant when we replace the y-column with the constant column (Dy):

scssCopy code
Dy = | 1  9 -1 |
     | 2  13  3 |
     | 3 -1 -5 |

Dy = 1 * ((13) * (-5) - 3 * 9) - 9 * (2 * (-5) - 3 * 3) + (-1) * (2 * 13 - 3 * 3)
Dy = 1 * (-65 - 27) - 9 * (-10 - 9) - (-1) * (26 - 9)
Dy = -92 - 9 + 17
Dy = -84

Step 5: Find the determinant when we replace the z-column with the constant column (Dz):

scssCopy code
Dz = | 1  2  9 |
     | 2 -1  13 |
     | 3  2 -1 |

Dz = 1 * ((-1) * (-1) - 13 * 2) - 2 * ((2 * (-1) - 13 * 3) + 9 * 3) + 9 * (2 * 2 - (-1) * 2)
Dz = 1 * (1 - 26) - 2 * ((-2 - 39) + 27) + 9 * (4 + 2)
Dz = 1 - 26 + 2 + 78 + 54
Dz = 109

Step 6: Finally, find the values of x, y, and z:

makefileCopy code
x = Dx / D = 122 / 30 ≈ 4.067
y = Dy / D = -84 / 30 ≈ -2.800
z = Dz / D = 109 / 30 ≈ 3.633

So the solution to the system of equations is approximately:

Copy code
x ≈ 4.067
y ≈ -2.800
z ≈ 3.633