Texas Instruments Voyage 200 Manual

2/7/02 ©2001, 2002 Texas Instruments
TI-89/TI-92 Plus/Voyage™ 200
Simultaneous Equation Solver
How To…
………
Start the Application Store Values
Enter Data Navigate
Solve a Problem Change Format
Examples
3Equations, 3 Unknowns Overdetermined Matrix
Undefined Variable Underdetermined Matrix
More Information
Customer Support Error Messages
TI

TI-89 / TI-92 Plus / Voyage™ 200 PLT Simultaneous Equation Solver App Page 2
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What Is Simultaneous Equation Solver?
Simultaneous Equation Solver finds solutions to systems of
linear equations. It provides a convenient, easy-to-use interface
to simplify setting up a problem, solving it, and exploring the
solution. You can:
•Enter, load, and edit coefficient matrices and up to five
vectors representing different right sides.
•Store the solution as a vector (an nx1 matrix).
•Display solutions as a list of values or in reduced row
echelon form (RREF).
•Solve over-determined systems of equations (systems in
which there are more equations than unknowns).
•Solve under-determined systems of equations (systems in
which there are fewer equations than unknowns).
•Solve systems containing both numeric and symbolic
coefficients.
•Identify whether a given system has a unique solution, an
infinite number of solutions, or no solution.

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Entering Data
Entering data manually
1. Start Simultaneous Equation Solver and select New.
2. Enter the number of equations and press D.Values entered
in this dialog box must be integers from 2 to 30.
3. Enter the number of unknowns.
4. Press ¸ ¸ to accept the problem size and display
the matrix. The first coefficient is selected, and its matrix row
and column designators are displayed on the entry line.

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5. Enter the first coefficient and press ¸.The cursor returns
to the matrix and moves to the next column (right). You can
change the cursor to move to the next row (down) instead in
the FORMATS dialog box.
You can enter a real or complex value or an expression that
resolves to a real or complex value. You can also enter
symbolic data.
Loading existing data
You can load an existing matrix as all or part of the problem from
within Simultaneous Equation Solver by using either of the
following methods.
•Press ƒ1:Load Augmented A|b.
•Press „1:Augmented A|b,2:Coefficient A,3:Constant b,or
4:Two Matrices A and b.

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Editing coefficients
1. Move the cursor to a coefficient and press ¸.The cursor
moves to the entry line.
2. Edit the value and press ¸.The new value appears in
the matrix.
Clearing coefficients
Press ƒ8:Clear All.
Starting a new problem
Press ƒ3:New Problem.

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Solving
1. Enter values for the coefficient matrix and values for the right
side. You can enter b1...b5and then select a different
right side.
2. Press ‡to solve the problem. The busy indicator on the
status line displays while the solution is calculated. The
solution values x1. . . xnare displayed in the specified
Exact/Approx mode. The RREF of the augmented matrix can
be shown instead.
If Angle Format is Degree and Complex Format is Polar,
computations are automatically performed numerically.
Displayed as list Displayed as RREF

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Storing
Once you have entered the coefficients, you can store:
•The augmented matrix (coefficients and the current right
side).
•The coefficient matrix.
•Constant vectors b1...b5.
•An individual element of a coefficient matrix or constant
vectors.
Once you have calculated the solution, you can also store:
•The solution as a vector (an nx1 matrix).
•An individual element of a solution.
•The RREF matrix.
•The residual vector, provided Show Residuals has been set
to YES in the FORMATS dialog box.

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From either the Coefficients screen or Solution screen, press …
and select
•1:Augmented A|b,select a folder, and enter a name. You
can also do this from ƒ.
•2:Coefficient A,select a folder, and enter a matrix name.
•3:Constant b,select a folder, enter a name and indicate
which right side contents (vectors b1...b5)you want to
store. The values are stored as a vector (an nx1 matrix).
•4:Solution x,select a folder, and enter a name. The
values are stored as a vector (an nx1 matrix).
•5:RREF Matrix,select a folder, and enter a name.
•6:Residuals,select a folder, and enter a name. The
values are stored as a vector (an nx1 matrix).
To store individual coefficients or roots from the Coefficients or
Solution screen:
1. Place the cursor on the value you want to store.
2. Press §.
3. Enter a variable name, and then press ¸.

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Navigating Within Simultaneous Equation
•¸moves the cursor to the next coefficient in the
equation or to the next equation, unless you change the
Cursor Movement direction in the FORMATS dialog box.
•Pressing ¸after entering b1iautomatically moves the
cursor to the next ai,1.You can alter the Cursor Movement
direction in the FORMATS dialog box.
•†returns to the Coefficients screen from the Solution
screen.
•A,B,C,or Dmoves among the coefficients for editing.
•2 A or 2 B on the entry line moves the cursor to the
beginning or end of the value.
•¤Aor ¤Bon the entry line highlights characters for
copying.
•¥ A,¥ B,¥ C,or ¥ D moves the cursor to the beginning
or end of the row or column.
•2 A,2 B,2 C,or 2 D in the matrix moves the
cursor one page.
•Use the 2 ˆ (TI-89) or ˆmenu to insert a column or
row, delete a column or row, or resize the matrix.

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•If the Split Screen mode is set to Top-Bottom when the
application is started, the TI-89 automatically switches to
Full;the TI-92 Plus / Voyage™ 200 PLT automatically places
the application in the top portion.
•To see the solution displayed in RREF, press ‡from the
Solution screen.
•If another application is running, 2 a toggles between
the applications.
Right sides
•To enter additional right sides (b2through b5), use the cursor
to move to those cells.
•To select a different right side to use in a calculation, use the
2 ‰ (TI-89) or ‰menu.
•…1:Augmented A
bstores only the current b. To store
additional right sides separately, use the …3:Constant b
command.

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FORMATS Dialog ox
To display the FORMATS dialog box, press ƒ9:Format or
shortcut keys ¥Í(TI-89) or ¥F.
Format settings
Column Width: Sets the number of characters to display in each
column. The default is 6. (Do not confuse this option with the
MODE setting Display Digits, which determines the number of
digits displayed.)
Cursor Movement: Determines whether the cursor moves to the
next column (right) or next row (down) after data are entered.
The default is Right.
Answer in RREF: Sets the default Solution screen display mode
when the system is solved the first time. If NO,the solution is
displayed in list form. If YES,the solution is displayed in RREF
form. The default is No.

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Show Residuals: Toggles display of the residual vector b-Ax,
where x is the computed solution. The default is No.
When Show Residuals is set to YES,residuals are displayed in
the column to the right of b5on the Coefficients screen. |resid| is
the magnitude of the difference between the coefficient matrix
times the computed solution and the right side. (For example,
residi=ai,1¦x1+ai,2¦x2…ai,n¦xn-bi,where i=1…n.) The entry line
shows the current residual element.
Tolerance: Default sets the tolerance to
5E-14¦max(dim(Augmented matrix)) ¦rownorm(Augmented matrix).
User-Defined enables the Tolerance field, where you can enter a
preferred tolerance level.

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Example — Three Equations, Three Unknowns
1. Press 3 „ and set Exact/Approx to Exact.
2. Start Simultaneous Equation Solver and select New.
3. Press
3D3¸ ¸ for a problem with three equations
and three unknowns.
4. Enter the coefficients:
a1,1 =9a1,2 = 8a1,3 = 7b11=2
a2,1 = 5a2,2 = M6a2,3 = M4b12=2
a3,1 = 1a3,2 = 5a3,3 = 9b13=7
5. Press ‡to solve the problem. The busy indicator on the
status line displays while the solution is calculated.
The values for x1, x2,and x3are displayed:
x1=72à481
x2=M547à481
x3=670à481

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Example — Undefined Variable
1. Press 3 „ and set Exact/Approx to Exact.
2. Start Simultaneous Equation Solver and select New.
3. Press
3D3¸ ¸ for a problem with three equations
and three unknowns.
4. Enter the coefficients:
a1,1 = 2a1,2 = 3a1,3 = M2b11=M3
a2,1 = 3a2,2 =M1a2,3 =unk b12=M2
a3,1 = 3a3,2 = 2a3,3 = M1b13=M1
5. Press ‡to solve the problem according to the mode
setting. The busy indicator on the status line displays while
the solution is calculated.
The values for x1, x2,and x3are displayed:
x1=26à(5…(5…unkN7))+3/5
x2=M104à(5…(5…unkN7)) N7à5
x3=M26à(5…unkN7)

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6. Press 3 „ and set Exact/Approx to Approximate.
The solution is automatically recalculated and expressed as:
x1=1.04à(unkN1.4)+.6
x2=M4.16à(unkN1.4) N1.4
x3=M5.2à(unkN1.4)

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Example — Overdetermined Matrix
1. Press 3 „ and set Exact/Approx to Exact.
2. Start Simultaneous Equation Solver and select New.
3. Press
5D3¸ ¸ for a problem with five equations
and three unknowns.
4. Enter the coefficients:
a1,1 = 9a1,2 = M6a1,3 = 2b11=6
a2,1 = M9a2,2 = 9a2,3 = M4b12=M1
a3,1 = M9a3,2 = 4a3,3 = M9b13=M7
a4,1 = M4a4,2 =M4a4,3 = 9b14=7
a5,1 = 1a5,2 = M9a5,3 = M7b15=8
5. Press ‡to solve the problem. The busy indicator on the
status line displays while the solution is calculated.
The display shows
No solution found.

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6. Press ‡again to view the RREF matrix.

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Example — Underdetermined Matrix
1. Press 3 „ and set Exact/Approx to Exact.
2. Start Simultaneous Equation Solver and select New.
3. Press
3D5¸ ¸ for a problem with three equations
and five unknowns.
4. Enter the coefficients:
a1,1 = 3a1,2 = 8a1,3 = 3a1,4 = 1a1,5 = 4b11=9
a2,1 = M1a2,2 = M7a2,3 = M2a2,4 = 7a2,5 = 4b12=4
a3,1 = M9a3,2 = M9a3,3 = M8a3,4 = 2a3,5 = 2b13=M2
5. Press ‡to solve the problem. The busy indicator on the
status line displays while the solution is calculated.

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The display shows two arbitrary constants (@1 and @2)and
an infinite number of solutions expressed in terms of those
constants:
x1=L307…@1à32N155…@2à16+15
x2=L37…@1à32N29…@2à16+3
x3=395…@1à32+211…@2à16N20
x4=@1
x5=@2