Solving Systems of Equations Using Mathcad
Charles Nippert
This set of notes is written to help you learn how to solve simultaneous equations using
Mathcad. You will solve a system of 2 simultaneous linear equations using successive
approximations or by using the symbolic processor. The methods you will use can be
easily adapted to other systems of equations.
For this quick tour you will solve the following system for F and T when B is 6.
F=T+B
0.75F = 0.3T + 0.95B
The first equation represents a total material balance around a process. The second
equation represents a material balance of one of the species in the streams
Mathcad has two convenient procedures for solving systems of equations. One method
uses symbolic manipulation (rearranging the equations and eliminating terms as you
would do if you were to solve the equations with pencil and paper). The other method
uses successive approximations (making a series of better guesses until the answers are
“close enough”).
Solving A System of Equations By Symbolically
1. Open Mathcad as you do normally. Begin by entering the value of the constant,
B.. Move the red cross to a spot about ½ inch below the menu bar and about ½
inch from the left hand side of the window on a normal sized screen using either
the arrow keys or the mouse cursor. Next type “B” (remember to use upper case
letters. Then choose “View/Toolbars/Evaluation” from the menu and select the
button (alternately, use the shortcut key “:”. Finally, enter the value 6. Your
screen should resemble figure 1.
Figure 1
After Step 1
Writing the equation
2. The equations that you are going to enter will be part of a "solve block". This
block isolates the equations and other information that Mathcad is going to use in
a particular operation from the rest of the information on the worksheet. The start
of the block is always marked by the word “given” which is directly typed onto
the worksheet. The word "given" is not part of any comment or text that the
program ignores but rather one of the statements Mathcad uses when processing
your equations. In most versions of Mathcad, comments appear in Arial font and
Mathcad statements such as "given" appear Times New Roman. Press “enter”
and type the word "given" below the equation you just entered. The word will
appear inside the rectangular box as shown in figure 2
Figure 2
“Given” marks the beginning of the solve block
3. Now enter the first equation, F = T + B. Press enter and type “F” just below the
word “given”. Next choose “View/Toolbars/Boolean” from the menu to open the
Boolean toolbar. Click the
button in the Boolean toolbar to create the equals
symbol. Type “T + B” using the keyboard. Press enter and enter the second
equation. Remember to use the “*” key to indicate multiplication. Your screen
should look like Figure 3 after you entered both equations.
Figure 3
After Entering Both Equations
4. You will now mark the end of the solve block and tell Matchcad which of the
variables (F,B or T) it should attempt to calculate. You will also tell Mathcad to
use the symbolic processor to find a solution. Both of these instructions are built
into one statement that uses the mnemonic “find” which is easy to remember
because you want the program to find a solution.
Finish the solve block by moving mouse cursor directly below the equation and
button on
typing "find (F, T) → ". You can create the arrow operator using the
the Evaluation toolbar. When you have completed the expression press "Enter".
The solution should appear to the right of the arrow. Your screen should look like
Figure 4. The top value (8 2/3) is the value of the first variable (F) and the second
value (2 2/3) is the value of the second variable (T).
Figure 4
Find the Solution to a System of Equations
Solving A System of Equations By Successive Approximations
The method of successive approximations starts with guesses values for each unknown,
then using an algorithm or set of rules to improving those guesses until the guesses
become “good enough”. The set up for solving a system of equations by successive
approximations is very similar to the setup for solving equations by symbolic
manipulation. The major differences are that initial guesses of the unknowns must be
specified and the
button instead of the
button on the evaluation toolbar. To
emphasize how similar the setups are, you will modify the solution from the previous
setup to use the numerical method of successive approximations.
5. Move the red cross cursor to a spot above the word given. Enter the values F= 5
and T = 10. Use the
button on the Evaluation toolbar or the “:” key to create
the assignment symbol in both expressions. Your worksheet should now look like
Figure 5
Figure 5
Enter the First Guesses of the Unknown Variables
6. Now, replace the arrow (used to solve symbolically) Move the mouse cursor over
the matrix with the answers and click the left moose button. A blue cursor will
appear as shown in Figure 6
Figure 6
The Blue Cursor Appears in the Solution Matrix
7. Now drag the mouse while holding the left mouse button until the entire matrix is
shown in inverted colors (black background and white letters) as shown below
Figure 7
Highlight the Matrix
8. Press the “Delete” key. The matrix and the arrow should disappear. Type “=” or
use the
button and the solution will appear as shown in Figure 8
Figure 8
The Solution Using Successive Approximations
Tips on Using Each Method
Each method has its own advantages and disadvantages. Successive approximations
work best when good first guesses are used. Successive approximation methods may
have difficulty finding complex solutions unless the initial guesses are complex. On
the other hand, the symbolic solver will find multiple roots of quadratics but may
have difficulty with some nonlinear systems. Figure 9a illustrates how the symbolic
processor finds the complex roots of a quadratic
Figure 9a
Finding Complex Roots Using the Symbolic Processor
The successive approximation solution requires a complex initial guess to converge
as shown in Figures 9b and 9c
Figure 9b
Failure of Successive Approximations with a real Initial Value
Figure 9c
Successive Approximations Finds a Root
When a Complex Initial Value Is Used
To reset the criteria for successive approximations, select “Math/Options” from the
Menu toolbar. A dialog box opens as shown in Figure 1. Set the “Convergence
tolerance” to the desired value
Figure 10
The Math Options Dialog