30S Applied Math
Mr. Knight – Killarney School
Unit: Linear Programming
Lesson 3: Solving Systems Using Graphmatica
Solving Systems Using
Graphmatica
LP-L3 Objectives:
To solve systems of linear
inequalities using Graphmatica
Learning Outcome B-1
Slide 1
30S Applied Math
Mr. Knight – Killarney School
Unit: Linear Programming
Lesson 3: Solving Systems Using Graphmatica
Graphmatica is a graphing utility that will allow you to graph
equations and inequalities.
Graphing lines and Entering Functions
Open Graphmatica.
The top line, the Menu Bar, contains various menus, e.g., File,
Edit, Redraw, View, Labels, and so on.
Theory – Graphing lines and
Entering Functions
Slide 2
30S Applied Math
Mr. Knight – Killarney School
Unit: Linear Programming
Lesson 3: Solving Systems Using Graphmatica
The middle line, the Button Bar, contains various icons. For
example, the second icon is "Open" a file, the third icon is
"Save" a file, and the sixth icon (a yellow pencil) is "Draw a
graph."
The third line is an open space where equations can be
entered. This is called the Equation Editor.
Below the graph area is the Status Bar that displays
messages from the Help menu.
Theory – Graphing lines and
Entering Functions
Slide 3
30S Applied Math
Mr. Knight – Killarney School
Unit: Linear Programming
Lesson 3: Solving Systems Using Graphmatica
Change the background colour of the graph area:
Click on View from the Menu Bar.
Click on the submenu Colors… to select an appropriate colour
scheme.
Click on White, below the third grid to change the background
colour to white.
•Click on OK to close
the dialog box
Theory – Change the background
colour of the graph area
Slide 4
30S Applied Math
Mr. Knight – Killarney School
Unit: Linear Programming
Lesson 3: Solving Systems Using Graphmatica
Draw a graph:
Enter y = 3x + 4 in the Equation Editor . Hit Enter.
The graph of y = 3x + 4
shows in the graphing area.
Notice: the Status Bar below
the graph shows the equation
that has been graphed.
Theory – Draw a graph
Slide 5
30S Applied Math
Mr. Knight – Killarney School
Unit: Linear Programming
Lesson 3: Solving Systems Using Graphmatica
You may want to change the part of the graph that you see.
Click on View on the Menu Bar.
Click on the submenu Grid Range…
Adjust the view by entering new values for the horizontal Left:
and Right: and the vertical Bottom: and Top:. For example, use
-7 and 7 as the horizontal left and right, and use -4 and 8 as the
vertical bottom and top.
Click on OK to see the
result of your choices.
Theory – Change Viewing Area
Slide 6
30S Applied Math
Mr. Knight – Killarney School
Unit: Linear Programming
Lesson 3: Solving Systems Using Graphmatica
To graph a new equation
If you want to graph a new equation, simply edit the old one or
delete it and enter a new one.
Key in y = -2x + 1 into the Equation Editor.
Click Enter to see the graph.
The second graph is shown
along with the first.
Theory – To graph a new equation
Slide 7
30S Applied Math
Mr. Knight – Killarney School
Unit: Linear Programming
Lesson 3: Solving Systems Using Graphmatica
An equation can be entered in any form. It does not have to
have to be written in the form y = mx + b.
For example, enter 3x - 2y = 4 directly into the Equation
Editor.
Press Enter to see all three graphs.
You can delete the last equation.
Click on Redraw from the Menu Bar.
Click on Delete Equation from the sub-menu. The last
equation, which is still highlighted, will be deleted.
You can delete all equations and graphs.
Click on Redraw from the Menu Bar.
Click on Clear All from the sub-menu. All equations and graphs
will be deleted.
Theory – To graph a new equation
Slide 8
30S Applied Math
Mr. Knight – Killarney School
Unit: Linear Programming
Lesson 3: Solving Systems Using Graphmatica
Closing Graphmatica
You can close Graphmatica by clicking on File from the Menu
Bar. Then click on Exit from the submenu. Select No when the
Save Untitled dialog box opens.
You can also close Graphmatica by clicking on the
x in the upper right corner of the main Graphmatica
screen.
Theory – Closing Graphmatica
Slide 9
30S Applied Math
Mr. Knight – Killarney School
Unit: Linear Programming
Lesson 3: Solving Systems Using Graphmatica
Try This Exercise Yourself:
Graph the following equations on the same grid.
•
y = 4x + 5
•
x + 2y = 6
•
y=5
•
x = -2
•
x = 3y - 3
After graphing all the equations, change the Grid Range to Left
(-10), Right (15), Bottom (-5), and Top (10) as shown.
Practice
Slide 10
30S Applied Math
Mr. Knight – Killarney School
Unit: Linear Programming
Lesson 3: Solving Systems Using Graphmatica
Solution
(The graph below is shown with a different grid range than
specified above in order to have it fit on the screen.)
Solution
Slide 11
30S Applied Math
Mr. Knight – Killarney School
Unit: Linear Programming
Lesson 3: Solving Systems Using Graphmatica
Graphing Inequalities
Graphmatica can also graph
inequalities.
Example 1
Open Graphmatica.
Input y > 2x + 3 into the Equation Editor. Use the ">" symbol
above the "." symbol on your keyboard.
Press Enter to see the graph. This graph has a dotted line
because the expression does not have an equality (=).
Delete this graph by clicking on Redraw > Delete Equation
Theory – Graphing Inequalities
Slide 12
30S Applied Math
Mr. Knight – Killarney School
Unit: Linear Programming
Lesson 3: Solving Systems Using Graphmatica
Input y 3x - 2 as y <= 3x - 2 into the Equation Editor.
Press Enter to see the graph. The graph has a solid line
boundary because the equality (=) is included in the set. Verify
that the region is correct by using (0, 0) as the test point in the
inequality:
y < 3x - 2
0 < 3(0) - 2
0 < -2.
This is false and means
that (0, 0) is not part
of the region.
The graph reflects this decision by shading the side of the line
that does NOT include (0, 0).
Theory – Graphing Inequalities
Example 2
Slide 13
30S Applied Math
Mr. Knight – Killarney School
Unit: Linear Programming
Lesson 3: Solving Systems Using Graphmatica
Deselecting Equations and Inequalities
Enter y = -2x + 1 into the Equation Editor and show the graph.
Enter y = 0.5x - 1.5 into the Equation Editor and show the
graph.
Theory – Deselecting Equations
and Inequalities
Slide 14
30S Applied Math
Mr. Knight – Killarney School
Unit: Linear Programming
Lesson 3: Solving Systems Using Graphmatica
You can "turn off" one of the graphs in the following way.
Click on the down arrow located at the end of the Equation
Editor.
This will show a "history" of the equations that you have
entered.
Theory – Deselecting Equations
and Inequalities
Slide 15
30S Applied Math
Mr. Knight – Killarney School
Unit: Linear Programming
Lesson 3: Solving Systems Using Graphmatica
Click on the inequality y <= 3x - 2. This action highlights it and
puts into the Equation Editor line.
Click on Redraw from the Menu Bar. Then, click on Hide
Graph from the sub-menu. The graph of this inequality will be
deleted from the coordinate grid.
You can show a graph that is hidden by selecting the equation
from the "history" list in the Equation Editor. Then click on the
Draw icon or press the Enter key.
Theory – Deselecting Equations
and Inequalities
Slide 16
30S Applied Math
Mr. Knight – Killarney School
Unit: Linear Programming
Lesson 3: Solving Systems Using Graphmatica
Try This Exercise Yourself
Draw the following graphs in Graphmatica
1. -4  x - 6
2. 2x - 7y > 21
3. -5x - 2y 14
4. 3y + 2x > 15
Practice Graphing Inequalities
Slide 17
30S Applied Math
Mr. Knight – Killarney School
Unit: Linear Programming
Lesson 3: Solving Systems Using Graphmatica
-4  x - 6
2x - 7y > 21
-5x - 2y 14
3y + 2x > 15
Practice Graphing Inequalities
Slide 18
30S Applied Math
Mr. Knight – Killarney School
Unit: Linear Programming
Lesson 3: Solving Systems Using Graphmatica
Express each of the following regions as an inequality
y  3x - 4
Practice Graphing Inequalities
y  -1.5x + 3
Slide 19
30S Applied Math
Mr. Knight – Killarney School
Unit: Linear Programming
Lesson 3: Solving Systems Using Graphmatica
Solving Systems of Linear Inequalities
The solution of a system of inequalities is the common area or
intersection where two or more half-planes overlap.
Example 1
Graph the solution defined by the following inequalities: x + y >
-1 and y > x - 2. The intersection of the two half-planes is the
solution. The area where there is double shading is the
solution.
In this common region,
the conditions of both
inequalities are satisfied
simultaneously.
Theory – Solving Systems of
Linear Inequalities
Slide 20
30S Applied Math
Mr. Knight – Killarney School
Unit: Linear Programming
Lesson 3: Solving Systems Using Graphmatica
A system can consist of more than two inequalities.
Example 2
Graph the solution defined by the following inequalities: y  2 ,
y  x - 1 and y  -x - 5.
Solution
Each inequality represents
a half-plane, so there are
three half-planes. The
solution is the area that
contains shading from all
three inequalities and is
a triangular region.
Theory – Solving Systems of
Linear Inequalities
Slide 21