Tutorial for Solution-Type Interactive Tool
1. PURPOSE: The following tutorial is designed to allow students and faculty, with
minimal Microsoft EXCEL experience, the ability to create similar tools or adapt
the tools already created by Bart Stewart and Jeff Libby.
2. METHODOLOGY: This tutorial will discuss in detail the steps to create the tool
and will utilize “screen shots” from Microsoft EXCEL to show the menus needed
to create the various components of the interactive tool.
3. INTERACTIVE TOOL DEVELOPMENT:
a) In the solution-type interactive tool, we are developing a program to
visually show the student the three types of solutions to a system of two
equations. The tool is designed to show the graphical solution as well as
stating what type of solution (unique, infinite, or no solution) results.
Furthermore, it helps to reinforce the concept of reduce row echelon form
(RREF) by displaying the matrix solution for RREF and also including the
general form of the matrix solution for each type of solution.
b) The following are detailed steps in developing the tool:
1) The first step is to set up a system of two equations as seen below:
The equations are set-up with additional columns between the variables x
and y in order to allow follow-on manipulation of the values of the sliders.
The sliders are utilized to change the constant coefficients for each
variable. Now that the basic system is set-up, we can then insert the
sliders by utilizing the following commands: View, Toolbars, Forms.
These commands open the menu to choose what type of slider the user
desires.
The following screen shot shows the commands for opening the menu to
obtain the desired slider:
This is the slider
mechanism chosen for
this interactive tool.
With the slider chosen, we can now assign properties to the slider in order
to create the range of values for our constant coefficients. Cell “B2”
below will be used as the range for our slider in order to adjust the
constant coefficient in Cell “A2”.
Cell “B2” is
the range of
our slider
Right click on
slider and chose
Format Control.
Range of slider can go
from 0 to 30000. Must
manipulate in order to
get desired values.
Links slider to
desired cell for
incremental
counter.
Can increment sliders
by integer values only,
but can manipulate to
increment by desired
values.
This interactive tool utilizes constant coefficients from –10 to 10
(excluding zero). Zero is excluded as it creates problems in the RREF
computations. In order to create this range, we utilized a conditional
statement for cell “A2”. The conditional statement utilized allows the
range of values in “B2”, which are controlled by the slider from 0 to 20, to
be manipulated to the desired range of -10 to 10. The following shows
the conditional statement used:
The line reads, if B2-10=0
then the constant coefficient
is B2-9. Other wise, the
value is Cell “A2” equals
B2-10.
First command
eliminates problem
created by zero
“B2” refers to the cell that the
slider controls. Ten is
subtracted to get the desired
range for our coefficients.
The same process is done to create the constant coefficient in Cell “A5”
for the variable “x” in the second equation. The slider is now linked to
Cell “B5” and the conditional statement in Cell “A5” controls the desired
range of the constant coefficient.
Now that the constant coefficients for the “x” variable have been created,
Column B can be hidden so that the information is not seen by the user.
This is accomplished using the following commands:
These commands will
hide any desired columns
that you do not want the
user to see. Can easily
“unhide” at a later time
to adapt the file for other
use.
The following is a screen shot of Column “B” hidden. This allows one to
correlate the constant coefficient with the “x” variable.
Information in
Column “B”
hidden.
The same process is now conducted to create the constant coefficients for
the “y” variable in both equations. The process utilizes Columns D and E
in the same manner discussed above. This time Column E will contain the
information that will later be hidden from the user.
The following is the result of creating the sliders for the constant
coefficients for the “y” variable in each equation:
Each slider controls
the range of the
constant coefficient
in the cell directly
above each individual
slider.
The next step was to develop the values for the right hand side of the
equality. This process is similar to the development of the constant
coefficients.
In this process, Column “H” is controlled by the sliders in Column “G”.
The command in “G2” and “G5” are utilized to create the range of our
constants. The range for the constants is from –10 to 10 (zero does not
create a problem in this case as a result you do not have to use a
conditional statement).
A color scheme was now created to delineate between the two equations
for later use in creation of the graph. The following commands were
utilized to open the command bar to change the font color and provide
background fill colors:
Provides various background
fill colors.
Provides various font colors.
2) The next process is to create the interactive graphical solution. The first
step is to create a column of values for the “x” variable (in our model the
range for the “x” variable was –20 to 20). This column was utilized to
solve for “y” in each of the equations separately. (NOTE: We are not
solving the equations simultaneously at this point to get a solution) The
following screen shots show the columns of data built and equations used
to compute the data:
Command to increment the values
for “x” by 1 from –20 to 20
Command to solve for
“y” for the blue
equation.
Command to solve for
“y” for the red
equation.
Now that the data is created, the interactive graphical solution can be
developed. This is accomplished by first highlighting the data in columns
K-M and utilizing the graph toolbar:
Toolbar for Graphing Command
Data used
to create
the
graphical
interactive
tool.
Utilized Line
Graph and
sub-type
shown.
The next several screen shots take you through the development of the
graph:
Ensure the Data Range
includes all three columns of
data.
Prior to going to the next menu, ensure to enter the series information by
selecting “Series” at the top of this window. It will bring up the following
window to enter the required information:
The blue equation is
already done and
can be verified by
the name change in
this window.
Enter the name of the
series and ensure the
“Values” are correct.
Ensure to list the range
values for the “x” variable or
your values on your graph
will not correspond.
The next step is to assign the title and axis labels:
Assigns name to
graph and to the
different axes.
The last step in the “Chart Options” allows you to place the graph in any
location you desire:
Desired location identified
in this block.
For our model, we just placed it in the same working sheet. If the colors
of your lines are not what you desire, they can easily be changed by
selecting each line separately and choosing the color you desire. This is
done by selecting one line using the left mouse key and then using the
right mouse key to get to the appropriate menu.
Right mouse key to get this
menu to open and then
choose “Format Data
Series” to change color of
line.
One can also change the
source data and various other
options from this menu. You
can also highlight the entire
graph with the left mouse key
and then use the right mouse
key to change desired data.
Choose your desired
color. In this case we
would choose blue to
match our color
coding from earlier.
The final product is an interactive graph. The graphed line solutions are
linked to the sliders and will change as the user changes the constants in
the system of equations.
Graphical Solutions
150
50
Blue Eqn
0
16
10
4
-2
Red Eqn
-8
-50
-2
0
-1
4
y-values
100
-100
-150
x-values
3) The next process is to develop the Reduced Row Echelon Form matrix in
order to determine the type of solution: Unique, Infinite, or No solution.
Furthermore, if the solution is unique, the tool will determine the unique
“x” and “y” variables. This entails several conditional statements and is
the most difficult part of developing this interactive tool. Before
developing the RREF matrix, the model will determine the type of
solution using conditional statements.
The column below called “Common Points” is used to determine when the
“y” values are equal in both equations
Conditional statement
identifying if Column L is equal
to Column M. If they are equal
then TRUE, otherwise FALSE.
Number of common is counting the
number of TRUE statements from
previous column. This statement says to
increment the counter if column N has a
TRUE statement.
Ratio for Blue and Red are just the ratio’s
of the “x” constant and “y” constant. The
use of these will be discussed below.
With the above information, we are now able to develop a conditional
statement to determine the type of solution. The following screen shot
shows the conditional statement command to determine the type of
solution.
The command states: if all conditional statements are TRUE then we have infinite solutions (ie:
same line), otherwise check the ratios. Having the same ratios means that the constant coefficients
for the “x” and “y” variables are the same thus the only difference is a different y-intercept which
leads to parallel lines NO Solutions. If neither of these conditions are satisfied then there is a
Unique Solution.
Now that the type of solution has been determined, the model now
develops the RREF matrix to show the corresponding form and also
determines the actual values for “x” and “y” for a Unique Solution.
The first step in this process is only cosmetic (ie: creating the matrix
without any consideration of conditional statements). This was
accomplished using the commands Insert, Object, Equation Editor 3.0:
Select Insert and
then scroll down
to select object.
Select Microsoft
Equation 3.0 in order
to build outside
brackets of the matrix.
May have to
manipulate to desired
size.
Select Ok to bring up
the following toolbar
Select desired objects
for brackets
Now that the cosmetic matrix brackets are set, you are ready to build the
2x3 RREF matrix. The first column is the same for all three solutions and
is created by simply typing in a “1” and “0” into the appropriate cells. The
remaining four cells required conditional statements and will be explained
in detail below.
We will first discuss the cell in the first row, second column. This value is
determined by the following conditional statement:
The cell “Q10” is the dummy
location for the type of solution
from the above steps.
This conditional
statement assigns the
value of zero to this
cell if the solution is
unique. If it is not
unique, the value is
simply the quotient of
the constants for the
“y” and “x” variable.
Simply dividing
through the first
equation by the
constant associated
with the “x” variable.
This conditional statement allows the cell in the first row, second column
to take on the appropriate value for RREF.
The next cell to determine is the second row, second column. This value
is determined by the following conditional statement:
Again, Cell “Q10” is
referenced. If the solution is
unique then a “1” is assigned to
the second row, second
column. Otherwise, a “zero” is
assigned as required for the
correct form for an infinite
solution or “no” solution.
The next step is to determine the value for the first row, third column.
This step requires an extensive conditional statement for the unique
solution. Since this cell is the solution for the “x” variable, one must first
solve one equation for “y” and substitute into the second equation in order
to solve both equations simultaneously for “x”. This command is shown
below:
Once again this command references the cell “Q10”. If “Q10” is a unique solution then the “x” variable
that solves each equation simultaneously is given by the above statement. If the solution is not unique,
then the variable is determined by dividing the right hand side of the equation by the “x” variable constant.
The final part of the RREF matrix is the second row, third column-“y”
variable solution. Once again the equations must be solved
simultaneously. This is accomplished by solving the first equation for “y”
and utilizing the unique solution value for “x” to determine the unique
solution value for “y”. If the solution is not unique, then other conditional
statements are utilized for an Infinite Solution and “No” solution. The
Cell
“F16” is
following screen shot shows the conditional statement used for this cell.
the unique
“x”
variable
value.
Computed
by the
statement
shown on
page 13.
If “Q10” is “Infinite
Solutions” then this cell
will be assigned a value of
zero.
If “Q10” is “No Solution” then
this cell is simply the
difference between the right
hand side values .
If “Q10” is “Unique
Solution” then this cell will
be assigned the value
computed by the above
statement.
This completes the difficult part of setting up the interactive tool. The
remaining parts are purely cosmetic and are up to the users discretion. One
can hide information not needed to be seen, color code as desired and add in
other useful information. For our tool, we added in the standard RREF form
for each of the three types of solutions, rearranged the information into a user
friendly template, and added several labels to identify the results. When you
are all done your model may look like the following:
Prepared by MAJ Jeff Libby and CPT Bart Stewart, June 2002.