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Simultaneous Linear Equations
To gain access to this window, select the menu item File / Linear Equations.
Select the Order
First select the order, i.e. the number of simultaneous equations, in the range 1 to 10 by clicking on the spin buttons to the right of where the order is shown, or by directly entering it. For large numbers of equations, there are many boxes, so they may be rather small. You can change the width of the window by dragging either side. A higher resolution monitor screen also helps. If you have inadvertently chosen the incorrect order, and only realise this after entering a lot of data, there is no problem. Just save the data first (see below), then change the order.
Enter the Data
Having chosen the number of equations to be solved, you now enter the data for each. The equations are conceptually of the form:
A1.X1 + A2.X2 + A3.X3 + ... = C
where A1, A2, A3 etc. each represent an expression to be entered into the
appropriate box, and the constant C is entered in the last
box of the line.
The 'Tab' key will move the focus from the currently selected box to the next box in sequence, moving left to right and down to the next line at the end of a line. If the shift key is held down when the 'Tab' key is pressed, the focus moves only in a vertical direction. You can of course also click into any box.
When all the data has
been entered, click on the button marked 'Calc', or press the
'Enter' key. The solutions, i.e. the values of X1, X2, X3, etc.,
will appear in the boxes on the bottom line. These values are also stored as
the variables x1, x2, x3,
etc., so that they can be accessed later. The solutions can also be viewed in
the multiple results window, by clicking on the
button marked 'Show'.
Notice that what you enter in the boxes need not be just numbers – you can enter any valid expressions, including multiple calculations and variables (as well as constants and functions). These are always calculated starting from the top left corner, moving along to the end of the line, continuing at the start of the next line underneath, and proceeding like this until ending in the bottom right hand corner. It is only after all of these values have been calculated that the results are handed over to the equation-solving engine. If there is a problem in evaluating an expression, a message box will announce the error and the problem expression will be highlighted. This highlighting will disappear as soon as you alter any expression.
Open and Save
The 'Open...' and 'Save...' buttons allow you to
store the equations as a plain text file, and reload them later. The 'Notes'
button 
allows you to write some notes to be saved
along with the data, and to view any notes you wrote previously. If you try to
load the equations when the chosen order does not match, you will be given a
choice: you can either change the current order (so that your equations will
load in the same form as you saved them), or you can keep the current order. If
you choose the latter option, the loaded equations will be truncated to fit, or
unfilled values will be left blank, as appropriate. (This is handy if you want
to increase or reduce the order of a set of equations.)
Logging
When calculation logging is enabled in UltimaCalc's
main window, the equations and solutions are automatically recorded to the log
file when they are calculated. If logging is not enabled, a log file button 
is
shown. Clicking on this when it is highlighted will calculate the current
solution and log the data and solution.
Note
Solving a large number of simultaneous linear equations can put a strain on the accuracy that is attainable. This is one reason why UltimaCalc works to 128 bit precision (about 38 digits). Some mathematicians seem to take a delight in finding sets of simultaneous linear equations which, even though they certainly have exact solutions from a mathematical point of view, are so difficult to solve numerically that the results can be wildly incorrect when calculated with more typical precision.