# Section 1.1.2.3: Simple Ordered Sets of Equations

## Ordering the Incidence Table for a Set of Equations

The equations for a simple VLE problem can written as an incidence table as follows:

 Variable: P P*1 p1 P*2 x2 p2 Equation (1) x Equation (2) x x Equation (3) x Equation (4) x x x Equation (5) x Equation (6) x x x

It was noted that the equations could be solved in the order:

• (1), (3) and (5)
• (2), (4)
• (6)
Let us rewrite the table with the equation rows in that order.

 Variable: P P*1 p1 P*2 x2 p2 Equation (1) x Equation (3) x Equation (5) x Equation (2) x x Equation (4) x x x Equation (6) x x x

It should now be clear that no equation contains more than one variable other than those already solved for in the equation rows above it.

This is particularly obvious if we now also reorder the columns, each of which corresponds to a variable, into the sequence for which the variables will be solved, viz:

• P*1 , P*2 and x2 from (1), (3) and (5)
• p1from (2), and p2 from (4)
• P from (6)
The table now looks like this:

 Variable: P*1 P*2 x2 p1 p2 P Equation (1) x Equation (3) x Equation (5) x Equation (2) x x Equation (4) x x x Equation (6) x x x

This is described as being lower triangular as all the non-zero entries fall in the bottom left of the table.

The entries along the diagonal indicate the variables for which each equation is solved by rearrangement into a formula.

All variables below the diagonal have been solved for in earlier equations.