Section 3.2.2: Ordering Equations Solutions

1.
Unknowns: x₁, x₂, t, p
There are four unknowns and four equations, therefore the set may be solved.

Incidence Table

Equation	x1	x2	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	x1	x2	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) x₁ = -3/(t + t²)
(1) x₂ = 1 - x₁
(4) p = (-t x₂)/x₁

Incidence Table:

Equation	x1	x2	y1	y2
(1)	1	1	1	1
(2)	1
(3)	1		1
(4)	1	1	1

Rearranging and reordering gives the following incidence matrix:

Equation	x1	y1	x2	y2
(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) x₁ = -3
(3) y₁ = 1 - x₁
(4) x₂ = y₁/x₁
(1) y₂ = x₁ + x₂ - y₁

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 + cv²
(4) P = RT/v
(1) k = (10^{A + B/T})/P

When i = 1, 2 the equations are:

(1) f₁ = (1 + ) l₁
(2) f₂ = (1 + ) l₂
(3) = v₁/l₁
(4) = v₂/l₂
(5) / =
(6) l₁ = 10

unknowns: , l₁, , l₂, v₁, v₂
There are six unknowns and six equations, therefore the set may be solved. Incidence Table:

Equation		l1		l2	v1	v2
(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	l1		v1		l2	v2
(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) l₁ = 10
(1) = (f₁/l₁) - 1
(3) v₁ = l₁
(5) = /
(2) l₂ = f₂/(1 + )
(4) v₂ = l₂

Section 3.2.1: Ordering Equations Questions

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