Solve the simultaneous equation below using Cramers rule2𝑥 + 3𝑦 − 𝑧 = 13𝑥 + 5𝑦 + 2𝑧 = 8−𝑥 + 2 𝑦 + 3𝑧 = 1

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

Step 1: Write the coefficients of the variables and the constants in matrix form: | 2 3 -1 | | x | | 13 | | 1 5 2 | * | y | = | 8 | |-1 2 3 | | z | | 1 |

Step 2: Calculate the determinant of the coefficient matrix (D): D = | 2 3 -1 | | 1 5 2 | |-1 2 3 |

D = (2 * 5 * 3) + (3 * 2 * -1) + (-1 * 1 * 2) - (-1 * 5 * -1) - (2 * 2 * 3) - (3 * 1 * -1) D = 30 + (-6) + (-2) - 5 + 12 + 3 D = 32

Step 3: Calculate the determinant of the x-variable matrix (Dx): Dx = | 13 3 -1 | | 8 5 2 | | 1 2 3 |

Dx = (13 * 5 * 3) + (3 * 2 * 1) + (-1 * 8 * 2) - (-1 * 5 * 1) - (2 * 2 * 3) - (3 * 8 * 3) Dx = 195 + 6 - 16 - 5 + 12 - 72 Dx = 120

Step 4: Calculate the determinant of the y-variable matrix (Dy): Dy = | 2 13 -1 | | 1 8 2 | |-1 1 3 |

Dy = (2 * 8 * 3) + (13 * 1 * -1) + (-1 * 1 * 2) - (-1 * 8 * -1) - (13 * 1 * 3) - (2 * -1 * 3) Dy = 48 - 13 - 2 + 8 - 39 + 6 Dy = 8

Step 5: Calculate the determinant of the z-variable matrix (Dz): Dz = | 2 3 13 | | 1 5 8 | |-1 2 1 |

Dz = (2 * 5 * 1) + (3 * 8 * -1) + (13 * 1 * 2) - (13 * 5 * -1) - (2 * 8 * 1) - (3 * 1 * 1) Dz = 10 - 24 + 26 + 65 - 16 - 3 Dz = 58

Step 6: Calculate the values of x, y, and z: x = Dx / D = 120 / 32 = 3.75 y = Dy / D = 8 / 32 = 0.25 z = Dz / D = 58 / 32 = 1.8125

Therefore, the solution to the simultaneous equation is x = 3.75, y = 0.25, and z = 1.8125.