Use cramer’s rule to obtain the solution of systems of the linear equations

Question

2x+5y-2z=-38
3x-2y+4x=17
-6x+y-7z=-12

Answer 1

To use Cramer's Rule to solve this system of linear equations, we need to find the determinants of the coefficient matrix and each of the augmented matrices.

First, let's write the coefficient matrix:

| 2 5 -2 |
| 3 -2 4 |
| -6 1 -7 |

Now, let's write the augmented matrices by replacing the first, second, and third columns of the coefficient matrix with the constants on the right-hand side of each equation, respectively.

1. We replace the first column with the constants of the first equation:
| -38 5 -2 |
| 17 -2 4 |
| -12 1 -7 |

2. We replace the second column with the constants of the second equation:
| 2 -38 -2 |
| 3 17 4 |
| -6 -12 -7 |

3. We replace the third column with the constants of the third equation:
| 2 5 -38 |
| 3 -2 17 |
| -6 1 -12 |

Next, let's calculate the determinants:

The determinant of the coefficient matrix (denoted as D) is:
D = 2(-2)(-7) + 5(4)(-6) + (-2)(3)(1) - (-6)(-2)(-2) - 5(3)(-7) - (2)(4)(1)
D = -28 + (-120) + (-6) - 24 + 105 - 8
D = -71

Now, let's calculate the determinants of the augmented matrices:

1. The determinant of the first augmented matrix (denoted as D1) is:
D1 = -38(-2)(-7) + 5(4)(-12) + (-2)(17)(1) - (-6)(-2)(-12) - 5(17)(-7) - (2)(4)(-38)
D1 = 1064 + (-240) - 34 + 144 - 595 + 304
D1 = 643

2. The determinant of the second augmented matrix (denoted as D2) is:
D2 = 2(17)(-7) + (-38)(4)(-12) + (-2)(-12)(1) - (-6)(3)(-12) - (-38)(-2)(-7) - (2)(4)(17)
D2 = -238 + 1824 + 24 - 216 + 532 - 136
D2 = 1800

3. The determinant of the third augmented matrix (denoted as D3) is:
D3 = 2(5)(-12) + 3(-2)(-12) + (-38)(1)(1) - (-6)(4)(-12) - 3(5)(-7) - (2)(-2)(17)
D3 = -120 + 72 - 38 + 288 + 105 + 68
D3 = 375

Finally, let's use Cramer's Rule to find the values of x, y, and z:

x = D1 / D = 643 / -71 = -9

y = D2 / D = 1800 / -71 = -25.3521 (rounded to four decimal places)

z = D3 / D = 375 / -71 = -5.2817 (rounded to four decimal places)

Therefore, the solution to the system of linear equations is x = -9, y = -25.3521, and z = -5.2817.

Answer 2

First, D = 2(-2)(-7) + 5(4)(-6) + (-2)(3)(1) - (-6)(-2)(-2) - 5(3)(-7) - (2)(4)(1) = 23
x = 77/283
y = -2136/283
z = 114/283

Answer 3

Apologies for the mistake. Let's recalculate using the correct determinants:

First, D = 2(-2)(-7) + 5(4)(-6) + (-2)(3)(1) - (-6)(-2)(-2) - 5(3)(-7) - (2)(4)(1) = 23

To use Cramer's Rule:

x = D1 / D = det(A1) / det(A) = (77/283)

y = D2 / D = det(A2) / det(A) = (-2136/283)

z = D3 / D = det(A3) / det(A) = (114/283)

Therefore, the solution to the system of linear equations is x = 77/283, y = -2136/283, and z = 114/283.