Solve equations graphically
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How to solve linear equations in one variable from a graph.

This topic is part of the TCS FREE high school mathematics 'How-to Library', and will help you to use a graphical method to solve algebra equations accurately.
(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 algebra equations in one variable accurately by graphing. This method will often work even when you can't solve the equation using algebra.


An equation is a statement that two quantities are equivalent.

For example, this linear equation:

5(x - 4)  =  3x + 2

means that 5(x - 4) and  3x + 2 are exactly the same.

Only certain values of the variable 'x' will make the equation true. The process of finding out the variable value(s) that makes the equation true is called ‘solving’ the equation.

If the left and right sides of the equation are graphed separately, the graphs will intersect wherever the 'x' coordinate is a solution of the equation.

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:  'Solve equations graphically.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.


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 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  Split the equation into two halves

·Â Â Â Â Â Â Â Â  If another symbol is used, replace with 'x'

·Â Â Â Â Â Â Â Â  Split the equation into two halves at the equal sign. Write down two equations each starting with 'y =' followed by half of the original equation.


For example, taking our example equation:

5(x - 4)  =  3x + 2

From the left hand side, we have:  y = 5(x - 4)

From the right hand side, we have: y = 3x + 2


Step 3 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 = 5(x - 4)

·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 = 3x + 2

·Â Â Â Â Â Â Â Â  Press the Enter key


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

Step 4 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 graph, we must press F10 twice before the intersection point of the graphs is visible.


Step 5 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 each point where the graph lines intersect. (See diagram) 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.


·Â Â Â Â Â Â Â Â  Write down the 'x' coordinate of the intersection point(s). These are the solutions of the equation.

There is only one intersection point (11,35) between the graphs:

 y = 5(x - 4) and  y = 3x + 2

Therefore the equation: 5(x - 4)  =  3x + 2 has one solution at x = 11.


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


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