**1.**

Unknowns: x_{1}, x_{2}, t, p

There are four unknowns and four equations, therefore the set may be solved.

Incidence Table

Equation | x_{1} | x_{2} | t | p |

(1) | 1 | 1 | ||

(2) | 1 | 1 | ||

(3) | 1 | |||

(4) | 1 | 1 | 1 | 1 |

Rearranging and reordering gives the following incidence matrix:

Equation | t | x_{1} | x_{2} | p |

(3) | 1 | |||

(2) | 1 | 1 | ||

(1) | 1 | 1 | ||

(4) | 1 | 1 | 1 | 1 |

This matrix is indeed lower triangular and so the equations may be solved in the following order:

(3) t = 4/a

(2) x_{1} = -3/(t + t^{2})

(1) x_{2} = 1 - x_{1}

(4) p = (-t x_{2})/x_{1}

Unknowns: x

There are four equations and four unknowns, therefore the set may be solved.

Incidence Table:

Equation | x_{1} | x_{2} |
y_{1} | y_{2} |

(1) | 1 | 1 | 1 | 1 |

(2) | 1 | |||

(3) | 1 | 1 | ||

(4) | 1 | 1 | 1 |

Rearranging and reordering gives the following incidence matrix:

Equation | x_{1} | y_{1} |
x_{2} | y_{2} |

(2) | 1 | |||

(3) | 1 | 1 | ||

(4) | 1 | 1 | 1 | |

(1) | 1 | 1 | 1 | 1 |

This matrix is indeed lower triangular and so the equations may be solved in the following order:

(2) x_{1} = -3

(3) y_{1} = 1 - x_{1}

(4) x_{2} = y_{1}/x_{1}

(1) y_{2} = x_{1} + x_{2} - y_{1}

Unknowns: k, P, T, v,

There are five unknowns and five equations, therefore the set may be solved.

Incidence Table:

Equation | k | P | T | v | |

(1) | 1 | 1 | 1 | ||

(2) | 1 | 1 | |||

(3) | 1 | ||||

(4) | 1 | 1 | 1 | ||

(5) | 1 | 1 |

Rearranging and reordering gives the following incidence matrix:

Equation | v | T | P | k | |

(3) | 1 | ||||

(5) | 1 | 1 | |||

(2) | 1 | 1 | |||

(4) | 1 | 1 | 1 | ||

(1) | 1 | 1 | 1 |

This matrix is indeed lower triangular and so the equations may be solved in the following order:

(3) = 0.76

(5) v = 1/

(2) T = a + bv + cv^{2}

(4) P = RT/v

(1) k = (10^{A + B/T})/P

When i = 1, 2 the equations are:

(1) f_{1} = (1 + ) l_{1}

(2) f_{2} = (1 + ) l_{2}

(3) = v_{1}/l_{1}

(4) = v_{2}/l_{2}

(5) / =

(6) l_{1} = 10

unknowns: , l_{1}, ,
l_{2}, v_{1}, v_{2}

There are six unknowns and six equations, therefore the set may be solved.
Incidence Table:

Equation | l_{1} |
l_{2} | v_{1} |
v_{2} | ||

(1) | 1 | 1 | ||||

(2) | 1 | 1 | ||||

(3) | 1 | 1 | 1 | |||

(4) | 1 | 1 | 1 | |||

(5) | 1 | 1 | ||||

(6) | 1 |

Rearranging and reordering gives the following incidence matrix:

Equation | l_{1} | v_{1} | l_{2} | v_{2} | ||

(6) | 1 | |||||

(1) | 1 | 1 | ||||

(3) | 1 | 1 | 1 | |||

(5) | 1 | 1 | ||||

(2) | 1 | 1 | ||||

(4) | 1 | 1 | 1 |

This matrix is indeed lower triangular and so the equations may be solved in the following order:

(6) l_{1} = 10

(1) = (f_{1}/l_{1}) - 1

(3) v_{1} = l_{1}

(5) = /

(2) l_{2} = f_{2}/(1 + )

(4) v_{2} = l_{2}

Section 3.2.1: Ordering Equations Questions

Course Organiser Last Modified 2/9/00