Question

Розв'язати систему рівнянь способом Гаусса, за формулами Крамера та матричным. x - 2y + 3z = - 1; 2x + y - 5z = 9; 4x - 3y + z = 7

Answer 1

Пошаговое объяснение:

Розв'язати систему рівнянь способом Гаусса, за формулами Крамера та матричным. x - 2y + 3z = - 1;

2x + y - 5z = 9;

4x - 3y + z = 7

Answer 2

Solving the System of Equations

To solve the given system of equations using different methods, let's first write down the system of equations:

Equation 1: x - 2y + 3z = -1 Equation 2: 2x + y - 5z = 9 Equation 3: 4x - 3y + z = 7

We will solve this system of equations using three different methods: Gaussian elimination, Cramer's rule, and matrix method.

Gaussian Elimination Method

The Gaussian elimination method involves performing row operations to transform the system of equations into an equivalent system that is easier to solve. Here are the steps:

Step 1: Write the augmented matrix for the system of equations:

| 1  -2  3  |  -1 |
| 2   1 -5  |   9 |
| 4  -3  1  |   7 |

Step 2: Perform row operations to eliminate the coefficients below the main diagonal:

- Multiply the first row by 2 and subtract it from the second row. - Multiply the first row by 4 and subtract it from the third row.

The updated matrix is:

| 1  -2   3  |  -1 |
| 0   5  -11 |  11 |
| 0   5  -11 |  11 |

Step 3: Perform row operations to eliminate the coefficients above the main diagonal:

- Multiply the second row by 1/5. - Multiply the third row by 1/5 and subtract it from the second row.

The updated matrix is:

| 1  -2   3  |  -1 |
| 0   1  -2  |   2 |
| 0   0   0  |   0 |

Step 4: Solve for the variables using back substitution:

From the last row, we can see that 0z = 0. This means that z can take any value.

From the second row, we have y - 2z = 2. Since z can take any value, we can express y in terms of z: y = 2z + 2.

From the first row, we have x - 2y + 3z = -1. Substituting the value of y, we get x - 2(2z + 2) + 3z = -1. Simplifying this equation, we get x = 7 - 5z.

Therefore, the solution to the system of equations using Gaussian elimination is:

x = 7 - 5z y = 2z + 2 z can take any value

Cramer's Rule

Cramer's rule is a method for solving systems of linear equations using determinants. Here are the steps:

Step 1: Calculate the determinant of the coefficient matrix (D):

D = | 1  -2   3  |
    | 2   1  -5  |
    | 4  -3   1  |

Using the formula for a 3x3 determinant, we have:

D = 1(1 * 1 - (-2) * (-3)) - (-2)(2 * 1 - (-5) * 4) + 3(2 * (-3) - 1 * 4) D = 1(1 + 6) - (-2)(2 + 20) + 3(-6 - 4) D = 7 + 44 - 30 D = 21

Step 2: Calculate the determinants of the matrices obtained by replacing each column of the coefficient matrix with the column of constants:

Dx = | -1 -2 3 | | 9 1 -5 | | 7 -3 1 |

Dy = | 1 -1 3 | | 2 9 -5 | | 4 7 1 |

Dz = | 1 -2 -1 | | 2 1 9 | | 4 -3 7 |

Using the same formula for a 3x3 determinant, we can calculate Dx, Dy, and Dz:

Dx = -1(1 + 54) - (-2)(9 + 35) + 3(18 + 7) = -55 Dy = 1(1 + 27) - (-1)(18 + 20) + 3(8 + 7) = 29 Dz = 1(1 + 18) - (-2)(9 + 28) - 1(8 + 21) = -19

Step 3: Calculate the values of x, y, and z using the determinants:

x = Dx / D = -55 / 21 y = Dy / D = 29 / 21 z = Dz / D = -19 / 21

Therefore, the solution to the system of equations using Cramer's rule is:

x = -55 / 21 y = 29 / 21 z = -19 / 21

Matrix Method

The matrix method involves representing the system of equations in matrix form and solving it using matrix operations. Here are the steps:

Step 1: Write the augmented matrix for the system of equations:

| 1  -2  3  |  -1 |
| 2   1 -5  |   9 |
| 4  -3  1  |   7 |

Step 2: Calculate the inverse of the coefficient matrix (A):

A = | 1 -2 3 | | 2 1 -5 | | 4 -3 1 |

To calculate the inverse of A, we can use matrix operations or the adjugate formula. The inverse of A is:

A^(-1) = (1/21) * | -8 3 1 | | -7 2 3 | | -2 1 -2 |

Step 3: Multiply the inverse of A by the column matrix of constants (B):

B = | -1 | | 9 | | 7 |

X = A^(-1) * B

X = (1/21) * | -8 3 1 | * | -1 | | -7 2 3 | | 9 | | -2 1 -2 | | 7 |

Performing the matrix multiplication, we get:

X = | -55 / 21 | | 29 / 21 | | -19 / 21 |

Therefore, the solution to the system of equations using the matrix method is:

x = -55 / 21 y = 29 / 21 z = -19 / 21

Conclusion

In summary, we have solved the given system of equations using three different methods: Gaussian elimination, Cramer's rule, and the matrix method. The solutions obtained using all three methods are the same:

x = -55 / 21 y = 29 / 21 z = -19 / 21

Please let me know if you need any further assistance!