We solve this equation by rearranging it to the form:

` x ` = 3

This is also an equation, because it contains an `=' sign, but it is
additionally
a ** formula ** because it is written so as to enable an
unknown quantity, here ` x ` to be calculated ** explicitly. **
The original form of the equation did not have this property.

Not all single equations in single unknowns may be easily
rearranged to provide formulas. For example:

` x² - 3x - ` 4 = 0

can be rewritten as a formula, but not particularly easily, although
the form is a standard one. It also yields more than one possible value
for ` x ` which can sometimes complicate
matters.

` x³ - x² - 3x - ` 4 = 0

can be rewritten only as a complicated formula, and so is
seldom solved in this way.

` x ` - log ` x ` = 0

This cannot be rewritten as a formula and so must be solved by a trial and
error method.

In general, we can rewrite an algebraic equation as a formula only when one of the following is true:

- The equation is
**linear**in the unknown, i.e. the unknown appears only raised to the first power, and does not appear as the argument of a function such as log, sin or exp. - The equation is nonlinear in the unknown, but the unknown appears
only once. E.g. we can rearrange:

log`x`- 4 = 0

to:

`x`= exp(4)

This is a set of two equations in two unknowns (` x `
and ` y`) and can therefore, in principle, be solved for both
its unknowns.

** For any set of equations to be solvable, there must be exactly as
many unknowns as equations.
**

It is also fairly obvious how the above equations can be solved.
The first can be rearranged to give a formula for ` x`:

` x ` = 3

The second can be rearranged to give a formula for ` y`,
so that ** when x
is known, **

The multiequation/multivariable equivalent of the condition for solving
a single equation by rearrangement to a formula,
which we will describe as
** direct solution **, is in fact two conditions.
The first of these should be obvious from the above example.

**
For direct solution, it must be possible to rearrange the equations
so that a formula
can be written for every unknown.
**

The second condition is less obvious but may be illustrated by a case where
it is not satisfied. consider the two equations:

` y ` - 4 ` x ` + 1 = 0

` y³ ` - ` x² ` = 0

Since neither equation contains either ` x `
or ` y ` on its own, it is not possible to solve immediately
for either of the unknowns. To be able to do this it would be necessary
for one equation to contain only one of the unknowns,
it would not matter which,
and the other could contain both. In general, a set of ` n `
equations to be solved this way
must be ** triangular **, i.e. one must contain no more than one unknown,
another no more than two, another no more than 3 and so forth.

The occurrence of these must also be in the correct pattern so that it is
possible for the equations to be both ** rearranged** and ** reordered
** so that they can be written, for example, for a set of
` n` equations in
unknowns `x`_{1} to `x _{n}`:

`x`_{1} = constant expression

`x`_{2} = f(`x`_{1}, constants)

`x`_{3} = f(`x`_{1}, `x`_{2}, constants)

`x`_{4} =
f(`x`_{1}, `x`_{2}, `x`_{3}, constants)

.

.

.

`x`_{n} = f(`x`_{1} ... `x _{n}`, constants)

There are a number of computer algorithms for testing and reordering sets of equations for solution in this way. Most modern spreadsheet systems will employ them automatically. There are also algorithms which will carry out the rearrangement, but these are not widely available. When using a simple program or spreadsheet the user will thus need to carry out the rearrangement, which normally will involve checking that a suitable order for direct solution does in fact exist.

As two special cases, we have provided in the modelling laboratory a simple iterative solver for single algebraic equations (L1.1) and a standard solver for sets of linear equations (L1.2).

Next - Section 3.2: Simple Ordered Sets of
Equations

Return to Section 3 Index