That the good question to post !

Effectively, the approach to solve such a system is greatly dependant about what is **h** standing for.

As a general convention in mathematics and scientific notations, the range of the letters in the alphabet generally indicate what the objects are :

- Letters at the end of the alphabet (x,y,z,t) are consider to be unknown (or variable),

- Letters in the middle of the alphabet (m,n,k,…) are consider to be a parameter; it is not an unknown by the actual value may change from one instance of the problem to another one.

- Letters at the beginning of the alphabet are know values or object, but numéric or complet expression are not write down for convenience or clarity of the formulae.

## h is parameter

If **h** is a parameter, the system have to be consider has three equations of three unknow variables. It can be solve with the classic methods and the solution will be expressed as a function of the **h parameter**.

{ x + y/4 = 11 <=> { x + y/4 = 11 <=> { x = 5.h <=> { x = 5.h

{ x + z/5 = y/3 { x - y/3 + z/5 = 0 { y = 44 - 20.h { y = 44 - 20.h

{ x/5 = h { x/5 = h { z = 5.( 44 - 35.h )/3 { z = ( 220 - 175.h )/3

For each value of parameter **h** the system admit an unique solution that coordinates (x,y,z) = ( *5.h , 44-20.h , 5.(44-35.h)/3* ) are affine functions of **h**.

## h is an unkonwn

If **h** is a unknown, then the system is an under-dimensioned system of linear equations . In this case, there is only thre possible ways:

1. The system has infinitely many solutions.

2. The system has a single unique solution.

3. The system has no solution.

Because there is less variable than equation, we know that such a system admit an infinity number of solutions.

{ x + y/4 = 11 <=> { x + y/4 = 11

{ x + z/5 = y/3 { x - y/3 + z/5 = 0

{ x/5 = h { x/5 - h = 0

The corresponding augmented matrix of such a linear system is:

( 1 1/4 0 0 11 )

( 1 -1/3 1/5 0 0 )

( 1/5 0 0 -1 0 )

Which lead to the echelon form:

( 1 0 0 -5 0 )

( 0 1 0 20 44 )

( 0 0 1 175/3 220/3 )

This indicate that the system admit an infinity number of solution:

For every of real *t*, the following quadruplet is a solution of the system:

{ x = 5.t

{ y = 44. - 20.t

{ z = (220. - 175.t)/3

{ h = t

## Conclusion

As can be observe, in the both case, the resolution follow the same ways and leads to the same coefficients. That why very few documentation explain how to handle under/over dimensioned systems. The mechanical of the resolution is similar to good dimensioned or square system.

Only the meaning of the solution is different, the same figure may be interpreted as an unique solution of parameter **h** or an infinite set of solution due to unknown *h* variable !

{ x = 5.t versus { x = 5.h

{ y = 44. - 20.t { y = 44 - 20.h

{ z = (220. - 175.t)/3 { z = ( 220 - 175.h )/3

{ h = t

EDIT : Have corrected one HUDGE ERROR concerning the infinite set of solution. Only parameter *t* is need !

Parameter *s* always egals to 1 (other value don't fullfill iniital system ).

*Edited: 13 June 2012, 6:06 a.m. after one or more responses were posted*