[ Home ]

This topic is part of the TCS FREE high school mathematics 'How-to Library', and will help you to solve simultaneous equations by finding where their graphs intersect.    (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.

Theory:

When there is an ordered pair that satisfies two or more equations, then this ordered pair is a simultaneous solution of the equations.

For example, consider these two equations:

y = 2x - 1

y = -x + 5

When ‘x’ is 2, then ‘y’ is 3 for both:

y = 2×2 - 1 = 3

y = -2 + 5 = 3

The ordered pair (x,y) = (2,3) is said to satisfy both equations, and is called a simultaneous solution of the equations.

If we graph both equations together, the graph lines will coincide at the point (2,3). This means that simultaneous solutions of equations can be found by graphing the equations and finding where they intersect.

Graphing with pencil and paper is inaccurate, but the Maths Helper Plus software allows you to use this approach with high accuracy.

Method:

IMPORTANT: This topic assumes that you know how to graph equations in Maths Helper Plus. Find out how by completing the 'Easy Start' tutorial in Maths Helper Plus help. To view the tutorial, select the 'Tutorial' option from the 'Help' menu in Maths Helper Plus.

We will use the example described in the ‘theory’ section above to demonstrate the steps for using Maths Helper Plus to solve simultaneous equations graphically.

Step 1  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.

Step 2 Enter the two equations

NOTE: You can type powers by using the '^' character. Type x² as 'x^2', or x³ as 'x^3'.

·Â Â Â Â Â Â Â Â  Type your first equation: y = 2x - 1

·Â Â Â Â Â Â Â Â  Press the Enter key

·Â Â Â Â Â Â Â Â  Type the second equation: y = -x + 5

·Â Â Â Â Â Â Â Â  Press the Enter key

 

For the example equations, the graph now looks like this:

Step 3 Adjust the graph scale

Depending on the equations you entered, important parts of the graphs may lie outside of the default graph scale. You can check on this with a simple zoom operation:

 

·Â Â Â Â Â Â Â Â  Zoom out. Briefly press the F10 key one or more times until you are sure that all of the intersection points between the two graphs are visible.

 

·Â Â Â Â Â Â Â Â  Zoom in. If you have zoomed out too far, hold down ‘Shift’ while you press F10 to zoom back in.

 

For the example graphs, there is no need to adjust the graph scale settings.

 

Step 4 Find the intersection points of the graph lines

·Â Â Â Â Â Â Â Â  Select the intersection tool by clicking this button: on the 'math tools' toolbar.

 

The intersection tool dialog box will appear (see below) and the mouse cursor will have this shape: when the mouse is moved over the graph.

 

NOTE: If the intersection tool dialog box covers up part of your graph, you can move it. Point with the mouse to the title: 'Intersection Tool' at the top of the dialog box. Now click and drag to a new location.

 

·Â Â Â Â Â Â Â Â  Move the cursor close to the point where the graph lines intersect. (See diagram below) Click the left mouse button. If the intersection point was found, a black dot will appear on the graph, and the dialog box will display its coordinates.

 

Repeat for each intersection point between the two graphs.

Cancel the intersection tool by clicking the 'Cancel' button on the intersection tool dialog box.