How to solve systems
of linear equations.

This topic is part of the
TCS FREE high school mathematics 'Howto Library',
and will help you to solve systems of linear equations with any number
of unknown variables.
(See the index page
for a list of all available topics in the library.)Â To
make best use of this topic, you need to download the Maths Helper Plus software.
Click
here for instructions. 
This topic shows you how to solve systems of linear equations
with any number of unknowns.
Theory:
Consider the following linear equations:
Â Â 2x
+ 3y – z
= –7
Â Â Â Â Â Â –4y
+ 6z = 26
5x + 9y
+ 2z = –7
If these three straight line functions have a single intersection point at
the point (x,y,z), then these coordinates will make each of these equations
true. (x,y,z) is then called the solution of the system of equations. This solution
can be found using the steps outlined below.
Method:
1Â Prepare the equations
a. Include zeros
Each
equation must have one term for each variable. For two
unknowns, every equation must have an 'x' term and a 'y' term. For three
unknowns, every equation must have an 'x' term, a 'y' term and a 'z' term.
If this is not the case, write in any missing terms with a zero
coefficient:
Â Â Â 2x
+ 3y – z
= –7
Â 0x
– 4y + 6z
= 26
Â 5x + 9y
+ 2z = –7
b. Include ones
Each
term must have a coefficient. If a term is a letter with no number in
front, like: 'x' or 'y' or 'z', then write it with a coefficient of one:
Â 2x
+ 3y – 1z
= –7
Â 0x – 4y
+ 6z = 26
Â 5x + 9y
+ 2z = –7
2Â Start
with an empty Maths Helper Plus document
If you have just launched the
software then you already have an empty document, otherwise hold down ‘Ctrl’
while you briefly press the ‘N’ key.
Â
3Â
Load the 'Simultaneous Linear Equations.tpl' template file
a.
If another symbol is used, replace with 'x'
b.
While holding down a 'Ctrl' key, press 'M'
to display the 'use teMplate' dialog box.
c.
Choose the 'Simultaneous Linear Equations.tpl' template file, then
click the 'Open' button.
4Â Â
Display the Matrix Editor
a. Hold down Ctrl and press the 'T' key to
view all of the text view.
b. Double click on
the text view beside the matrix calculator data set, anywhere in the
area shaded red in the diagram below...
This will display the options box.Â
Click the 'display program' option check box to turn this option off...
d. Click the
'Matrix Editor' tab at the top of the dialog box to display the matrix
editor ...
5Â
Enter the coefficients
'Coefficients' are the numbers from the left side of the equations.
a. Make sure the 'Now editing' list box
indicates 'A' is being edited.Â
(If not, click the selection arrow and choose
'A')Â
b. Click on the editing window, then type the coefficients of the equations
separated by commas and on separate lines, like this:
2,
3, 1
0, –4, 6
5, 9, 2
6.
Enter the right side values
(These are the numbers on the right side of the equations.)
a. Select 'B' in the 'Now editing' list
box...
b. Click on the editing window, then type the numbers, one on each line, like
this:
–7
26
–7
7.
Solve the equations
a. Select 'X' in the 'Now editing' list
box.
b. Click the 'Run' button. The solutions will
appear in a vertical column, like this:
1
–2
3
So the solution: (x,y,z) = (1,2,3)Â
Â