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HP 50g Graphing Calculator - Solving Linear Systems of Equations using Matrices

The Numeric Solver

The HP 50g has a numeric solver that can find the solutions to many different types of problems. It is invoked by pressing the ORANGE shift key followed by the key, or .

When pressed, the CHOOSE box below is displayed:

Figure 1: Displaying the options in the numeric solver

The first choice allows for the solution of an equation containing a number of unknowns. The second choice solves differential equation problems. The third choice solves for zeroes of a polynomial. The fourth choice to solve linear systems of equations is the one of interest here. The fifth choice invokes the finance solver. The sixth choice begins the multiple equation solver. To select the linear systems solver, press . The 50g displays the following screen:

Figure 2: Selecting the linear systems solver

There are three input areas on this form. The first area is labeled A: and is where the matrix containing the coefficients of the system of linear equations are entered. The second area is labeled B: and is where the constants from the system of equations are entered. The third area is where the values of the unknown variables X in the systems of linear equations are returned when a solution is found. To enter the coefficients in a matrix, press when the cursor is beside the area labeled A: and the MatrixWriter is launched.

Figure 3: Entering the coefficients in a matrix

In many ways, this screen works like any spreadsheet. Enter numbers and they will go in the highlighted cell. The menu labels at the lower left corner of the screen, and , determine the direction the cursor moves after a data point has been entered, either right to the next column or down to the next row. In this example, the selection is to move right after each data point has been entered. This is indicated by the square present next to the menu label. To change the way the cursor will move, press the menu label for the direction desired and the square in the menu label will change accordingly. If a column is too small to show the data entered, the and menu keys may be used to expand or shrink the area displayed for each column.
Enter the first coefficient from the first equation by keying in the numbers and pressing the key. The cursor will move to the right into the second column where the second coefficient should be keyed with the key pressed to accept this value. At this point, the cursor will be in column 3. If the system of linear equations being entered has more than two unknown variables, continue entering the data until done. Use the and keys to move back to the first column and enter the coefficients from the second equation. Continue entering rows of data into the matrix until done. If at any time you notice a mistake in the data, use the arrow keys to go back to the incorrect data value, key in the correction, press the key to accept the change, and then use the arrow keys to go back to where you were. After entering a matrix, the screen would look something like this:

Figure 4: The screen displays after entering a matrix

To accept the data as input, press the key and the matrix will be returned to the system of linear equations solver.

Figure 5: Displaying the linear equations solver

To enter the constants for the linear equations, press the key and press the menu label above the key. The MatrixWriter will be brought up a second time to enter the constants.
NOTE: The constants are entered as a column of numbers rather than a row. Enter the numbers the same way that the coefficients were entered.

Figure 6: Entering the constants for the linear equations

When the key is pressed, the constants are returned to the linear equation system solver in the B: area, as shown below.

Figure 7: Displaying the constants in the B: area

To solve for the values of X that make the system of linear equations work, use the arrow keys to move down to the X: area. Press the menu label above the key and the values of X1 and X2 that solve the linear system are returned and displayed.

Figure 8: Displaying the constants in the X: area

The solution may be seen easier by pressing above the key to view the solution in the MatrixWriter. If the columns are too small to see many significant figures, press the menu label to make them wider.

Figure 9: Displaying the data by expanding the columns

You may also exit to the stack and view the solutions on the first level. To exit to the stack from within the MatrixWriter, as we were immediately above, press and the display will show:

Figure 10: Displaying the solution

Solving linear systems of equations

There are many times when a problem is presented of the form:

5 X1 + 10 X2 + 3 X3 = 20

4 X1 + 1 X2 + 8 X3 = 15

and the values of X1 , X2 , and X3 are needed. Manual methods to eliminate variables and reduce the number of unknowns used to be the only way to solve these. The 50g makes such solutions easy.

Practice solving linear systems

Example 1

What are the values of X1 , X2 , and X3 that solve the following system of linear equations?

5 X1 + 10 X2 + 3 X3 = 20

4 X1 + 1 X2 + 8 X3 = 15

Solution

To solve this linear system, use the numeric solver as shown.

Figure 11: Entering the equation using the numeric solver

Figure 12: Displaying the constants in the A: area

Figure 13: Entering the second portion of the equation

Figure 14: Displaying the values in the B: area

Figure 15: Displaying the X1 result

Figure 16: Displaying the data by expanding the column

Figure 17: Displaying the result

Answer

The value of X1 is 0.9788, the value of X2 is 1.1376, and the value of X3 is 1.2434, all approximately of course.

Example 2

What are the values of X1 , X2 , and X3 that solve the following system of linear equations?

3 X1 + 5 X2 + 1 X3 = 36

1 X1 + 2 X2 + 4 X3 = 42

4 X1 + 3 X2 + 2 X3 = 28

Solution

To solve this linear system, use the numeric solver as shown.

Figure 18: Entering the equation using the numeric solver

Figure 19: Displaying the values in the B: area

Figure 20: Displaying the X1 result

Figure 21: Displaying the result

Answer

The value of X1 is –1.9024, the value of X2 is 6.8293, and the value of X3 is 7.5609, all approximately.

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