Solve the following linear system for x using Cramer’s rule.

Show work.
x + 2y – 3z = -22
2x – 6y + 8z = 74
-x – 2y + 4z = 29

Cramer’s rule uses determinants to find the solution

Set up a 3by3 matrix using the coefficients of the x,y and z, and find its determinant, calling it D.

You can see a good example at
http://www.mathwords.com/c/cramers_rule.htm

Which method is most efficient for solving large systems, Cramer's Rule or Gaussian Elimination and why?

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

Step 1: Set up the matrix of coefficients.
We will create a 3x3 matrix using the coefficients of x, y, and z from the given equations.

| 1 2 -3 |
| 2 -6 8 |
| -1 -2 4 |

Step 2: Find the determinant of the matrix (D).
The determinant of the matrix D is found by using the formula:
D = (a(ei - fh) - b(di - fg) + c(dh - eg))
where a, b, c, d, e, f, g, and h represent the entries of the matrix.

D = (1*(-6*4)-(2*(2*4))+(-3*(-2*8)))
= (-24 - 16 + 48)
= 8

Step 3: Find the determinants of the matrices obtained by replacing the first column of the coefficient matrix with the constant values of the linear system.

Matrix D1:
| -22 2 -3 |
| 74 -6 8 |
| 29 -2 4 |

D1 = (-22*(-6*4)-(2*(74*4))+(-3*(74*(-2))))
= (528 - 296 - 444)
= (528 - 296 - 444)
= -212

Matrix D2:
| 1 -22 -3 |
| 2 74 8 |
| -1 29 4 |

D2 = (1*((74*4)-(8*29))-(2*((-1*4)-(4*29)))+(-3*((-1*8)-(74*4))))
= (296 - 58 + 180)
= 418

Matrix D3:
| 1 2 -22 |
| 2 -6 74 |
| -1 -2 29 |

D3 = (1*((-6*29)-(74*(-2)))-(2*((-1*29)-(74*1)))+(-22*((-1*(-6))-(2*2))))
= (-174 - 156 + 148)
= -182

Step 4: Calculate x, y, and z.
The values of x, y, and z can be obtained by dividing each determinant D1, D2, and D3 by the determinant D.

x = D1 / D = -212 / 8 = -26.5
y = D2 / D = 418 / 8 = 52.25
z = D3 / D = -182 / 8 = -22.75

Therefore, the solution to the linear system is x = -26.5, y = 52.25, and z = -22.75.