Section 3.2: Simple Ordered Sets of Equations

As has been shown, there are many cases when a set of algebraic equations in several unknowns can can be reordered and/or rearranged to enable all variables to be solved for in sequence rather than simultaneously.

We can create an incidence or occurrence table (or matrix) for the set of equations, see example (1.1.2.2) with a row for each equation and a column for each variable.

By trial and error, it is possible to try rearranging the rows and columns to see if a `lower triangular' structure can be arrived at, with all the non-zero elements (i.e. x's in the table) arranged into the bottom left hand half of the table.

Here are some examples to try (section 3.2.1).

If you are using a modern spreadsheet to solve a problem involving algebraic equations of this sort, then the spreadsheet will probably sort the equations out in this way using one of the standard algorithms which have been developed for this task. You will still have to perform some analysis of your equations to ensure that the individual equations are written, after rearrangement if necessary:

• as formulas (i.e. with one variable on the l.h.s.),
• with each variable appearing once and only once on the l.h.s. of a formula, and
• so that a suitable order can be found if it exist.
The last of these is quite a subtle requirement. If more than one rearrangement appears to be possible, use the incidence matrix to help determine an order.

If a correct order has not been determined, or does not exist, then the spreadsheet system will probably complain and give an error message about `circular references'.

If a lower triangular arrangement of the equations is not possible, then a iterative trail-and-error procedure must be used. This will be discussed after we have consider iterative solution of single equations in the next section.