Contents
_________________________________________________________
Chapter 1 : Introduction............................................................................................... 1
What is Algebrator ?........................................................................................................... 3
Installing Algebrator on your Computer............................................................. 4
System Requirements.................................................................................................................. 4
Installation................................................................................................................................... 4
Miscellaneous files...................................................................................................................... 6
Typographical Conventions............................................................................................. 7
Chapter 2 : Getting Started.......................................................................................... 9
Basic Workspace Expression Manipulation........................................................ 11
Solving Equations, Inequalities and Graphing.................................................. 23
Solving Systems of Equations, and Graphing..................................................... 33
Chapter 3 : Advanced Features................................................................................ 47
Using Visibility and Domain Options......................................................................... 49
Other Modes of Problem Solution............................................................................. 57
Chapter 4 : Teacher’s Tools...................................................................................... 65
Creating Documents and Workbooks..................................................................... 67
Generating Problems and Documents..................................................................... 77
Changing Supervisor Options......................................................................................... 85
Chapter 5 : Command Reference.............................................................................. 93
Working With Files............................................................................................................... 95
Opening an Expression File................................................................................................... 95
Saving Workspace.................................................................................................................... 96
Saving Workspace under a Different Name......................................................................... 96
Printing in Algebrator............................................................................................................. 96
Setting Printer Options......................................................................................................... 100
Exiting Algebrator................................................................................................................. 100
Basic Workspace manipulations.............................................................................. 102
Creating an Algebraic Expression...................................................................................... 102
Editing an Algebraic Expression........................................................................................ 102
Displaying an Expression..................................................................................................... 103
Deleting an Expression......................................................................................................... 103
Deleting All Workspace Expressions.................................................................................. 103
Generating Algebraic Expressions from a Template....................................................... 104
Simplifying Expressions................................................................................................... 105
Showing Next Simplification Step....................................................................................... 105
Showing Simplification Result............................................................................................. 105
Selecting Simplification Transformation........................................................................... 106
Combining Like Terms........................................................................................................... 107
Combining Like Factors....................................................................................................... 107
Multiplying Powers................................................................................................................ 107
Distributing a Power............................................................................................................. 108
Adding Fractions.................................................................................................................... 108
Multiplying Fractions........................................................................................................... 108
Dividing Fractions................................................................................................................. 109
Reducing a Fraction.............................................................................................................. 109
Rationalizing a Denominator.............................................................................................. 109
Performing One Factoring Step........................................................................................... 110
Complete Factoring............................................................................................................... 110
Expanding a Term.................................................................................................................. 110
Eliminating Parenthesis........................................................................................................ 111
Eliminating Special Cases.................................................................................................... 111
Reducing Order of a Radical................................................................................................ 111
Eliminating Negative Exponents......................................................................................... 112
Making a Base Positive......................................................................................................... 112
Making a Numerator or a Denominator Positive............................................................ 112
Converting a Decimal Number into a Fraction................................................................ 113
Converting a Fraction into a Decimal Number................................................................ 113
Converting Radical Expression into Exponential
Expression...................................... 113
Converting Exponential Expression into a Radical
Expression................................... 114
Rewriting an Expression in a Common Form................................................................... 114
Evaluating an Expression..................................................................................................... 114
Simplifying a Subexpression................................................................................................ 115
Finding the Greatest Common Factor................................................................................ 115
Finding the Least Common Multiple.................................................................................. 115
Manipulating an Expression................................................................................................ 116
Checking a Manipulation.................................................................................................... 116
Solving Equations and Inequalities....................................................................... 118
Showing Next Solution Step................................................................................................. 118
Showing the Solution............................................................................................................ 119
Checking the Solution........................................................................................................... 119
Selecting Solution Transformation..................................................................................... 120
Adding an Expression to Both Sides of an Equation or
Inequality.............................. 120
Subtracting Common Terms from Both Sides of an
Equation or Inequality............... 121
Subtracting an Expression from both sides of an
Equation or Inequality.................. 121
Multiplying Both Sides of an Equation or Inequality by
LCD...................................... 121
Multiplying Both Sides of an Equation or Inequality by
an Expression..................... 122
Dividing Both Sides of an Equation or Inequality by
Common Factors..................... 122
Dividing Both Sides of an Equation or Inequality by
the Variable’s Coefficient...... 122
Dividing Both Sides of an Equation or Inequality by an
Expression........................... 123
Exponentiating Both Sides of an Equation or Inequality
by a Power......................... 123
Making Bases Equal in an Exponential Equation.......................................................... 123
Using Quadratic Formula to Solve an Equation............................................................. 124
Creating Linear Equations / Inequalities from Higher
Order Equation / Inequality 124
Switching Left and Right Side of an Equation or
Inequality......................................... 125
Creating Equation or Inequality Solution........................................................................ 125
Deleting Extraneous Root from the Solution.................................................................... 125
Switching Top and Bottom Equation in a System of Two
Equations........................... 126
Adding Equations in a System of Two Equations............................................................. 126
Substituting a Selected Variable in a System of Two
Equations................................... 126
Solving one of the Equations in a System of Two Equations......................................... 127
Manipulating an Equation or Inequality.......................................................................... 127
Checking a Manipulation.................................................................................................... 128
Changing a Solution Variable............................................................................................ 128
Choosing a Solution Method for Solving a System of Two
Equation.......................... 129
Graphing.................................................................................................................................... 130
Showing Next Step in Conversion to Graphable Form................................................... 130
Showing Graphable Form.................................................................................................... 130
Graphing a Curve or a Solutions........................................................................................ 131
Explaining the Solution Process............................................................................... 132
Showing the First Transformation in the Current Step................................................... 132
Showing the Next Transformation in the Current Step................................................... 133
Showing the Previous Transformation in the Current
Step........................................... 133
Showing Context Sensitive Explanation for Current
Transformation......................... 134
Returning from Step Display................................................................................................ 134
Returning to Step Display..................................................................................................... 135
Displaying Term Definition.................................................................................................. 135
Using Supervisor Options................................................................................................ 136
Changing Transformation Visibility................................................................................... 136
Changing Visibility to None................................................................................................ 136
Changing Visibility to Low.................................................................................................. 136
Changing Visibility to Medium........................................................................................... 137
Changing Visibility to High................................................................................................. 137
Changing Visibility to All..................................................................................................... 137
Changing the Solution Domain........................................................................................... 137
Allowing Calculations with Real Numbers....................................................................... 138
Allowing Calculations with Complex Numbers............................................................... 138
Allowing Use of Quadratic Formula................................................................................... 139
Setting a Number of Steps to Keep...................................................................................... 139
Entering Supervisor Password............................................................................................ 140
Changing Supervisor Password.......................................................................................... 141
Enabling and Disabling Menu Access............................................................................... 141
Changing Default Directories.............................................................................................. 144
Changing Background Color.............................................................................................. 144
Changing Print Colors.......................................................................................................... 145
Black and White Printing..................................................................................................... 145
Color Graph Printing............................................................................................................ 145
Color Graph and Explanation Printing............................................................................ 146
Opening an Option File........................................................................................................ 147
Saving an Option File........................................................................................................... 147
Saving an Option File under a Different Name................................................................ 147
Creating and Generating Documents.................................................................... 148
Creating a Test or Homework Document........................................................................... 148
Creating a Workbook............................................................................................................ 151
Generating
Documents......................................................................................................... 154
Editing a Document............................................................................................................... 154
Accessing Help....................................................................................................................... 156
Displaying Help Contents..................................................................................................... 156
Displaying Help Search Dialog........................................................................................... 156
Accessing Algebrator Tutors................................................................................................ 156
Displaying About Box............................................................................................................ 157
Editing Expressions............................................................................................................. 159
Cutting a Subexpression in Algebraic Editor................................................................... 159
Copying a Subexpression in Algebraic Editor Input Line............................................. 159
Pasting a Subexpression in Algebraic Editor................................................................... 160
Clearing Input Line in Algebraic Editor........................................................................... 160
Deleting a Subexpression in Algebraic Editor................................................................. 160
Undoing Changes in Algebraic Editor.............................................................................. 161
Selecting Expressions....................................................................................................... 162
Copying a Subexpression in Algebraic Two-dimensional
Display.............................. 162
Replacing a Subexpression in Algebraic Two-dimensional
Display........................... 162
Substituting a Subexpression in Algebraic
Two-Dimensional Display....................... 163
Importing Expressions...................................................................................................... 164
Inserting an Existing Expression in Algebraic Editor.................................................... 164
Inserting a Solution Form in Algebraic Editor................................................................ 164
Creating Templates........................................................................................................... 166
Defining a Template in Algebraic Editor........................................................................... 166
Clearing a Template in Algebraic Editor.......................................................................... 167
Exiting Algebraic Editor................................................................................................ 168
Exiting Algebraic Editor....................................................................................................... 168
Canceling Algebraic Editor................................................................................................. 168
Renaming Expression in Algebraic Editor........................................................................ 169
Saving and Closing Documents.................................................................................. 171
Saving a Document................................................................................................................ 171
Saving a Document under Different Name......................................................................... 171
Closing a Document............................................................................................................... 172
Editing Problems.................................................................................................................. 173
Cutting a Problem.................................................................................................................. 173
Pasting a Problem.................................................................................................................. 173
Copying a Problem................................................................................................................ 174
Clearing a Problem................................................................................................................ 174
Deleting a Range of Problem............................................................................................... 174
Inserting Problems............................................................................................................. 176
Inserting a Problem................................................................................................................ 176
Inserting a File........................................................................................................................ 176
Moving between Problems............................................................................................. 177
Moving to the Next Problem................................................................................................. 177
Moving to the Previous Problem......................................................................................... 177
Moving to the Last Problem................................................................................................. 178
Moving to the First Problem................................................................................................ 178
Moving to a Specific Problem.............................................................................................. 178
Creating Answers............................................................................................................... 179
Generating Problem Answers............................................................................................... 179
Duplicating Problem Answers............................................................................................. 180
Miscellaneous non-Menu Operations.................................................................... 181
Focusing an Expression........................................................................................................ 181
Scrolling through Workspace Expressions....................................................................... 181
Selecting a Subexpression on Algebraic Editor Input
Line........................................... 182
Selecting a Subexpression in Algebraic Editor
Two-dimensional Display................ 182
Entering an Expression in Select Transformation Editor............................................... 183
Chapter 6 : Appendices................................................................................................. 185
Algebrator’s Limitations............................................................................................. 187
Valid Characters in Algebraic Editor.................................................................. 189
Workspace and Expressions......................................................................................... 191
Expression Workspace Defined............................................................................................ 191
Expression Defined................................................................................................................. 192
Types of Workspace Expressions......................................................................................... 192
Step Expressions..................................................................................................................... 193
Graphable Expression........................................................................................................... 193
Group of Expressions............................................................................................................. 194
Solution Check Expression.................................................................................................. 194
Solution Expression............................................................................................................... 194
Template Expression.............................................................................................................. 195
Instance Expression............................................................................................................... 195
Ambiguous Expressions in Algebraic Editor.................................................................... 195
Common Form of an Algebraic Expression....................................................................... 196
Expression Evaluation.......................................................................................................... 196
Files and Documents......................................................................................................... 197
Expression File....................................................................................................................... 197
Option File............................................................................................................................... 197
Document File......................................................................................................................... 198
Workbook Definition File..................................................................................................... 198
Test or Homework Document............................................................................................... 198
Workbook Document............................................................................................................. 199
Template Document................................................................................................................ 199
Instance Document................................................................................................................. 199
Solution Process.................................................................................................................. 201
Solution Process Defined...................................................................................................... 201
Transformation........................................................................................................................ 201
Special Cases Transformations............................................................................................ 202
Visibility................................................................................................................................... 202
Next Step.................................................................................................................................. 202
Next Step in Select Transformation..................................................................................... 203
Expression Manipulation Modes........................................................................................ 203
Context Sensitive Explanations........................................................................................... 204
Error, Warning and Information Messages...................................................... 207
License Agreement.............................................................................................................. 231
Index............................................................................................................................................ 235
_________________________________________________________
This section will give you a basic overview
of Algebrator software. It will also
lead you through the installation process and explain typographical conventions
used in this manual.
What
is Algebrator ?
Installing
Algebrator on Your Computer
Typographical
Conventions
_______________________________________________________________________________________________________________________________________________________________________
Algebrator is a computer algebra
system specifically designed to teach pre-college algebra. It is meant to be
used students as well as teachers. Algebrator is not dependent on predefined,
‘hardwired’ problems. A user can enter and manipulate a wide variety of
symbolic expressions found in a typical algebra textbook. Specifically, Algebrator
covers the following areas of algebra :
·
Simplification
of rational, radical and complex expressions
·
Solving
linear and higher order equations and systems of two linear equations
·
Graphing
conic sections and equation/inequality solutions
If you are a student...
You can enter a problem from your algebra
practice workbook, and Algebrator will solve it for you, using natural solving strategy; the problem solution will be displayed in
the same fashion a teacher would write it on a blackboard - step by step. If
there is a step you don’t understand, explanation is just a mouse click away.
Algebrator doesn’t only give you a general rule; it also relates it to a particular problem at
hand. If you would like to see how Algebrator does this, go to Getting Started chapter now.
If you are a teacher...
If you install Algebrator in your school PC
lab, you can control how your students use it. For example, you could disable Result Only option to force step by step solution mode.
Algebrator also lets you design your own problem templates that can be used to
generate a large number of similar problems. Finally, Algebrator is a powerful tool for creating
printable test, homework and workbook documents (it will even automatically generate
correct and incorrect multiple choice
answers!). To familiarize yourself with basic operations, go to Getting Started section now. More
advanced control options are described in Advanced
Features section. Supervisor controls and document creation topics can be
found in Teacher’s Tools section.
_______________________________________________________________________________________________________________________________________________________________________
Algebrator
will work on any IBM-PC compatible microcomputer running Windows 3.1 or Windows
95. Currently, Algebrator does not work under Windows NT.
To install
Algebrator software:
From a diskette :
·
place
the enclosed diskette in drive a:
·
choose File | Run (Win
3.1) or Start
| Run (Win95)
·
type a:\setup
·
press <Enter>
·
follow
on-screen instructions
From a CD
(assuming that d:
drive is CD ROM drive) :
·
place
the enclosed CD in CD ROM drive d:
·
If
Autorun (Win95) is enabled, startup
process will start automatically.
·
If
Autorun is not enabled, choose :
·
Start | Run (Win95)
or
·
File | Run (Win
3.1)
·
type d:\setup
·
press <Enter>
·
follow
on-screen instructions
Notice that
Algebrator window contains three icons. If you want to :
·
start
Algebrator, double-click on ![]()
·
start
Algebrator help only, double-click on ![]()
·
remove
Algebrator from your system, double-click on ![]()
If you are
running Windows 3.1 and installing Algebrator from a CD-ROM, on an older CD-ROM
drive, you might get a message “Error writing directory” at the very end of the
installation process. This message can usually be ignored. Click on <OK> and start Algebrator. If the problem persists
(i.e. you get some other kind of error message while trying to run the
program), reinstall the software from the diskette.
Beside
the Algebrator program, an extensive collection of problem, template and
document files has also been installed. Assuming that Algebrator is placed in c:\alg, these files can be found in \problem and \doc subdirectories.
Algebrator
is shipped with no supervisor password. We strongly recommend that you create
your own password as soon as possible. In case you ever forget it, delete the
option file that Algebrator is currently using. This will delete the password.
For security reasons it is recommended that only supervisors have write access
to a disk containing option files.
_______________________________________________________________________________________________________________________________________________________________________
In
this manual, the following fonts and styles have specific meanings :
Example Meaning
Open | File Arial
(‘|’ delimited) . Specifies a sequence of
menu items. Usually these menu items have to be ‘clicked’ for a
particular operation to be performed.
<Enter> Arial (‘<>‘ enclosed). Specifies a
control key to be pressed.
test.doc Arial. Specifies expression and file names
x+y^2 Arial bold. Specifies a string of characters
that has to be typed (usually within algebraic editor).
double-click on ... Book
Antiqua, bold. This typeface is used in tutor chapters. It describes a specific
action that user needs to take.
edit
mode Book Antiqua, italic. This typeface is used
for terms that have specific meaning in Algebrator environment
_________________________________________________________
Getting
Started section is designed
to lead you through some basic Algebrator operations. Step by step tutors found
in the following chapters can also be accessed on-line by choosing Help | Tutors.
Basic
Workspace Expression Manipulation
Solving
Equations, Inequalities and Graphing
Solving Systems of
Equations and More Graphing
Tutor
1
_______________________________________________________________________________________________________________________________________________________________________
When you first start Algebrator by double-clicking on
your
screen should look like this :

We
will refer to this main Algebrator window as workspace. You can think of it as of an electronic
blackboard. So, how do we write problems on our blackboard? In order to create
an algebraic expression, we need to invoke a special algebraic editor. Let’s do that by either choosing Workspace
| New or by clicking on
button. The
following editor window should now appear on your screen :

Notice that algebraic editor consists of two
parts :
·
input
line (one
line input box right underneath the buttons)
·
two-dimensional
display (larger window bellow the input line)
Input line is used to enter an expression in
‘calculator’ notation. For example, a over b would be entered as
a/b and a
squared would be entered as a^2. As an expression is being entered, it will also appear in its
standard ‘book’ form on the two dimensional expression display. By glancing at
this display, you can immediately see whether your entry is correct.
Go ahead and enter the following expression in algebraic editor input line :
(x^2-y^4)/(x+yy)
Your screen should now look like this :

Input line cursor is green informing you that the editor
is in check mode. In this mode characters are checked for
correct algebraic syntax as soon as they are entered. If an error is detected
the cursor turns red, algebraic editor
goes into error mode and
displays appropriate error message.
Try entering ++ at the end of the current expression to see how this works. Delete both pluses before proceeding. If you move the cursor to the left
(‘inside’ the expression), algebraic editor goes into edit mode and
stops performing the syntax check, so that you can freely edit the expression.
You can always get back to check mode by pressing <Enter> or <End> key. In edit mode the cursor is yellow.
Here are some of the more important features
of algebraic editor :
·
If two
variables are entered in sequence and editor is in check mode, multiplication sign will
be inserted between the variables.
·
All
open parenthesis can be closed by ] (parenthesis fill character) at any time.
·
If you
want an existing workspace expression to become a part of the expression that
you are editing, enter its full name in
caps.
·
Blank
space is not a valid character. For a complete list of valid characters see Appendix B.
Two-dimensional display always displays a
completed algebraic expression. This means :
·
When
algebraic editor expects another operand (i.e. in expression ’a+’ ), a ‘dummy’ operand (?)
will be inserted in two-dimensional display, wherever the character is
expected.
·
When
algebraic editor expects the closing parenthesis (i.e. in expression ‘(a+b’ ) a closing parenthesis will be displayed in two-dimensional
display, even if user still hasn’t
entered it on the input line.
Let’s go on with defining our first
expression. If you are in edit mode (yellow cursor) press <Enter> key once to go to check mode
(green cursor). Once you are in check mode press <Enter> key
one more time to let
Algebraic Editor know you are done. The following dialog box should now be
displayed on your screen.

Enter the following name : myfirst.
Expression that you entered has now been placed at the bottom of workspace
window. Its full name is myfirst.exp
.
Using the same procedure
enter expression :
(x+y)^2/(x+y)
and name it mysecond.
Your screen should now look like this :

Notice that mysecond.exp is now displayed in dark red color, while myfirst.exp is black. This means that mysecond.exp is focused. Transformations such as simplification or
graphing are always applied to the focused expression. Practice changing expression focus by
clicking the right mouse button on the expression that you wish to focus. Notice that when you position the mouse
over a particular expression, its name will be displayed in the status line at
the bottom of the screen.
It is possible that some large workspace
expressions will not entirely fit on the screen. In order to improve overall
clarity, scroll bars for each individual expression in workspace are not
provided. When you need to see the whole expression, click on it, to bring up
the algebraic editor
and then use the scroll bars.
You will now learn how to combine the
existing expressions into a single one. First, click left mouse button anywhere on myfirst.exp
window to bring it into
algebraic editor. Then, type
: +MYSECOND.EXP at the end of the expression. Make sure that you use uppercase letters. If
you don’t, Algebrator will try to parse string mysecond.exp as an algebraic expression instead of
interpreting it as an expression name. Press <Enter> key twice to exit the editor. If you did everything
correctly, the expression at the bottom of the screen (myfirst.exp) should now look like this :

In order to save the workspace (all three
expressions), click
on
on main button bar and enter myfile as
a file name.
Now, let’s see if this file really got
saved. First, clear the workspace by choosing Workspace |
Clear All menu item. Then, click on
to
retrieve the file that you have just
saved. Enter myfile as a file name. The screen should now look
exactly the same as it did before the workspace got cleared.
OK, let’s do some real work now. Why did we
create this expression in the first place ? To have it simplified, of course.
Algebrator can simplify expressions in a variety of different ways. Right now,
let’s assume that we are only interested in the result of simplification. Click on
to create the simplified form of myfirst.exp.
Your screen should now look like this :

Lots of computer algebra systems (including
some calculators) can do what Algebrator just did. The real power of Algebrator
is in the ability to solve a problem in a step by step fashion and provide
context sensitive explanations. That’s what we are going to do next.
It is usually a good idea to get rid of old
solution, before resolving a problem. So, let’s do that first : press <Del> key, select myfirst@1.exp and then press <Enter>
key. The solution is now
gone.
Make sure that myfirst.exp is focused. Click on
on main button bar to perform the first step
toward simplifying expression myfirst.exp.
Notice that
button on main button
bar is no longer disabled. This means that context sensitive explanation is
available for this step.
Click on
on main button bar to
highlight the first transformation in step myfirst@1.exp. Click on
on main button bar to
display explanation window for the first transformation. Your screen should now
look like this :

Notice that in addition to displaying the
difference of two squares formula, Algebrator also tells you how that formula
relates to the problem at hand (A = x, B = y, M = 2).
If you click on any green term in explanation window, its definition will be provided. You can
remove the definition window by clicking anywhere on
the screen. To remove the explanation window from the screen click on
.
Notice that
and
buttons are enabled.
This means that there are more explanations available for this step. You can click on
if you want to see the
next transformation. Practice displaying different
transformations and explanations.
Now, let’s exit step display. First, make
sure that no explanation windows are displayed. Then, click on
to return to standard
workspace expression display.
Use
and
buttons to see all the step of the
simplification process for myfirst.exp. You
will know that you are done when a message ‘Further
simplification is not possible’ appears on the screen. At this point
expression myfirst.exp is completely simplified.
Your screen should now look like this :

Notice that only the last three steps are
kept in workspace. This is done so that you don’t have to manage an excessive
number of expression. Number of steps kept in workspace can be changed by
choosing Options | Keep Steps menu
item.
Your workspace now contains a number of
expressions. Practice
moving through workspace by using the following
browse buttons :
Scrolls
to the ‘top’ of expression space. The first workspace expression will be
focused and displayed at the bottom of the screen.
Scrolls
to the ‘bottom’ of expression space. The last workspace expression will be
focused and displayed at the bottom of the screen.
Scrolls
‘up’ by one expression. Expression focus does not change.
Scrolls
‘down’ by one expression. Expression focus does not change.
Now, let’s save the workspace expressions
into a new file. Choose File |
Save As
and name the file : myfile2.
Notice that, if you had clicked on
instead, the previous
contents of myfile would have been overwritten by the current
workspace contents.
_______________________________________________________________________
Congratulations ! You have now learned how to perform basic
Algebrator operations.
Tutor
2
_______________________________________________________________________________________________________________________________________________________________________
In this tutoring session you will learn how
to solve some simple equations and inequalities. You will also discover how to
represent solutions graphically.
After you start Algebrator, enter the following equation in Algebraic
Editor :
2x+6=0
and name it firsteq. Click on
button
couple of times to see how Algebrator solves this equation. If you need
explanations for any step, click on
button. Message ‘Further solution is not possible’ will
appear on the screen, once the problem is completely solved.
Your screen should now look like this:

Notice that the equation name has extension sol. When graphing, this extension tells Algebrator to use a number line instead
of (x,y) coordinate system. So, let’s see the graphical representation of this
solution.
Click on
to obtain the following graph :

To remove the solution graph from the screen
double-click on graph
window close button.
Now we are going to see what happens when we
try to solve an equation containing a variable other than
x. Click on
button to create another equation. Instead of
typing the entire equation, we will import firsteq into the editor. You probably remember from
the first tutoring session that this can be done by entering equation name in
upper case letters. There is an even easier method; simply choose Quick
Pick | Expression from algebraic editor menu and select firsteq.equ from the list of existing expressions.
Then, click on
button to import firsteq.equ into algebraic editor.
Now, replace x in
2*x+6=0 by
y . Exit
the editor and
name the new equation secondeq .
Click on
button on main button
bar to see the first step in the solution process.
A message ‘Can not solve equations of this type’ should now be displayed on
your screen. Press
<Enter> key to acknowledge the message. This message has appeared because
Algebrator tried to solve the equation for variable x , which doesn’t exist any more. As you can see, it is important
to know which variable the equation is being solved for, before the solution
process is started.
OK, so how do we change the solution
variable? It is really very simple; choose
Solve | Solve for menu
item from the main menu bar and then select the appropriate variable (y). Try solving the equation again. This time,
there should be no problems.
_______________________________________________________________________
Now it’s time for more complicated
equations. First, clear
the workspace. Then type in the following equation :
x^2-(x-1)/(x-1)=0
and name it nonlin.
Make sure that x is the solution variable. Let Algebrator solve this equation step
by step. Look at explanations if any steps are unclear.
By the time ‘Further solution is not possible’ message is displayed, your
screen should look like this :

Are we done now ? The answer is no ! Since the original equation contains the
variable x in the denominator, the
solutions {1,1} need to be checked.
Click on
button on main button
bar to start the solution check process. The first solution has now been
substituted for variable x in the
original equation. Notice that the
focused check equation is named nonlin@1.ck1 (first step in checking the first
solution).
Click on
button on main button
bar again to see the next step in solution check process . The messages ‘Division by zero’ and ‘Original solution not valid’ will by
displayed on your screen.
Good thing we decided to check our solution!
‘1’ turns out not to be a valid solution. Acknowledge the message(s) by pressing the <Enter>
key.
Click on
button couple of times
to check the second solution. After Algebrator determines the validity of this
solution, the solution set will be updated. Your screen should now look like
this :

_______________________________________________________________________
So far we have solved equations only. Now
let’s try some inequalities. Clear the workspace and type in the
following inequality :
x^2+5x+6>0.
Name it nonlin.
Make sure that x is the solution
variable.
Click on
button on main button
bar several times to see how Algebrator solves this inequality. If you need
explanations for any step, click on
button.
After a few steps, messages ‘Only
some of the following intervals are valid solutions’ and ‘To get valid solution, perform check
solution operation’ will appear on the screen. Acknowledge them by pressing the <Enter>
key. Your screen should now
look like this :

Notice that focused expression name is nonlin?.sol .The ‘?’ indicates that solution intervals
have not been checked.
Click on
button on main button
bar to start the process of eliminating invalid solution intervals. A value from the leftmost solution
interval (-4) has now been substituted for variable x in the original inequality.
Keep
clicking on
button on main button
bar until a series of messages appears on the screen informing you about the
validity of the solution intervals. The correct solution should now be
displayed at the bottom of your screen :

Inequality solutions can also be graphed. Click on
button on main button bar to see the graph :

To remove the solution graph from the screen
double-click on graph
window close button.
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Congratulations ! You have now learned how
to solve equations and inequalities and how to graph their solutions.
Tutor
3
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In this tutoring session you will learn how
to solve and graph systems of two linear equations. You will also find out how
to graph some non-linear curves.
Enter
the following system of equations in algebraic editor :
2x+3y=0,5y-4x=7
Notice that equations must be separated with
a ‘,’ (comma) character. No blank spaces are allowed. Name the system linsys. Notice extension attached to the system’s
name is set .
Make sure that (x,y) are the solution variables. Click on
button on main button
bar several times to see how Algebrator solves this system of equations. If you
need explanations for any step, click on
button.
By the time ‘Further solution is not possible’ message is displayed, your
screen should look like this :

Click on
to
obtain the following solution graph :

To remove the solution graph from the screen
double-click on graph
window close button.
You probably remember from your classroom
lecture that there are several methods that we can use to solve a system of
equations. One employed in the above example is called the elimination method. Now, we will tell Algebrator to solve the same system by
using the substitution method.
First, delete all expressions except for the original linsys
system. Then choose Solve
| Method | Substitution to
change the system solution method from elimination to substitution. Repeat the solution process and notice the differences. The final
result, however, should be equivalent to
the one obtained earlier.
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So far, we have seen how Algebrator solves
systems of linear equations. How about non-linear system ?
Start algebraic editor and choose Quick Pick | Expression to import system.set
into the editor. Change
the expression so that it looks like this :
2x^2+3y=0,5y^2-4x=7
Instead of pressing <Enter> key (and saving the system as linsys), choose Exit | Rename and enter parabolas as the expression name.
Try solving this system. A message : ‘Can not solve non-linear system’. Algebrator
can solve this kind of system only graphically. So let’s try graphing.
Another message : ‘This expression is not
in graphable form’ is displayed.
What exactly is a ‘graphable form’ ? Here is a table that shows what
Algebrator considers graphable :
________________________________________________________________
Curve Graphable form
________________________________________________________________
line y = ax + b
vertical
line x = a
vertical
parabola y = (x + a) 2 + b
horizontal
parabola x = (y + a) 2 + b
vertical
hyperbola (x + a) 2
(y + b) 2
¾¾¾¾ - ¾¾¾¾
= 1
m n
horizontal hyperbola - (x + a) 2 (y + b) 2
¾¾¾-¾ + ¾¾¾¾ = 1
m n
circle (x + a) 2 + (y +
b) 2 = r
ellipse (x + a) 2 (y + b) 2
¾¾¾¾