# Section 3.5.4: Solutions to Example Questions

1. The incidence table for the equation set is:

 Equation x1 x2 x3 x4 x5 (1) 1 1 1 (2) 1 1 (3) 1 1 1 (4) 1 1 (5) 1

Rearranging and reordering the incidence table leaves us with:
 Equation x5 x1 x3 x4 x2 (5) 1 (4) 1 1 (3) 1 1 (1) 1 1 1 (2) 1 1
For this set of equations, equation (5) is the head, equations (4), (3) and (1) the partition, and equation (2) the tail. Either of x1, x3 or x4 would be suitable for tear variables here.

2. The incidence table for the equation set is:

 Equation x1 x2 x3 x4 (1) 1 1 1 1 (2) 1 1 1 1 (3) 1 1 (4) 1 1

Rearranging and reordering the incidence table leaves us with:
 Equation x4 x3 x1 x2 (4) 1 1 (3) 1 1 (1) 1 1 1 1 (2) 1 1 1 1

In this case there is no head or tail. Instead all of the equations form the partitions. Either of x2, x3 or x4 would be suitable for the tear variable here. x1 would not be suitable as estimating a value for x1 does not leave an equation with only one further unknown present.

3. The incidence table for the equation set is:
 Equation x1 x2 x3 x4 (1) 1 1 (2) 1 1 1 (3) 1 1 1 1 (4) 1 1

Rearranging and reordering the incidence table leaves us with:
 Equation x1 x2 x3 x4 (4) 1 1 (1) 1 1 (2) 1 1 1 (3) 1 1 1 1

Equations (4), (1) and (2) form the partition here. This leaves equation (3) to be the tail. There is no head. Either of x1, x2 or x3 would be suitable choices for the tear variable.