Systems of Linear Equations
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How to solve systems of linear equations.

This topic is part of the TCS FREE high school mathematics 'How-to 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.) 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.


Consider the following linear equations:

  2x + 3yz = –7
      –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.

Download the free support file... We have created a Maths Helper Plus document containing the completed example from this topic. You can use this to practice the steps described below, and as a starting point for solving your own problems.

File name:  'Systems of linear equations.mhp'   File size: 3kb
Click here
to download the file.

If you choose 'Open this file from its current location', then Maths Helper Plus should open the document immediately. If not, try the other option: 'Save this file to disk', then run Maths Helper Plus and choose the 'Open' command from the 'File' menu. Locate the saved file and open it. If you do not yet have Maths Helper Plus installed on your computer, click here for instructions.


1  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:

   2x + 3yz = –7
x – 4y + 6z = 26
 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:

  2x + 3y1z = –7
 0x – 4y + 6z = 26
 5x + 9y + 2z = –7

2  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.


3  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.

4   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. 
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 ...

5  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. 
(If not, click the selection arrow and choose 'A') 

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. 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:


So the solution: (x,y,z) = (1,-2,3) 


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