Before we start using the calculator, note that these equations are given in standard form. That is, they appear as Ax + By = C, where A, B, and C are numeric values. For two equations, in two unknowns (variables) x and y, we could write the equations in a general standard form as:
A | B | C |
D | E | F |
a_{1,1} | a_{1,2} | b_{1} |
a_{2,1} | a_{2,2} | b_{2} |
The problem that we were given was:
a_{1,1} is 3 | x_{1} is x | a_{1,2} is 4 | x_{2} is y | b_{1} is 14 |
a_{2,1} is 5 | x_{1} is x | a_{2,2} is -7 | x_{2} is y | b_{2} is -45 |
With all of that out of the way, we are finally ready to start using the calculator. The steps shown before assume that the calculator is turned on, that we are not in any menu, and that the screen is clear.
| In Figure 1 we enter the desired matrix directly from the calculator keyboard. To do this we use the to generate a [ to signal the start of the matrix. A second produces a [ to signal the start of a row. Then we give the coeeficients and the constant of the row, separated by commas. The keys for our example are . Then we use and to produce ][ to signal the end of the first row and the start of the second row, respectively. The values in the second row are . Then, we need to end the second row with a ], and end the matrix with a ]. We do this with the keys and . Finally, we want to store this matrix on the calculator. We start to store it by pressing the key. |
| On a TI-83 or a TI-83 Plus we need to store a matrix into one of the 10 predefined matrices, [A], [B], [C], [D], [E], [F], [G], [H], [I], or [J]. To do this we need to find the name of the desired matrix. The names are available in the MATRIX menu. On the TI-83 we press the key to open the MATRIX menu. (As noted at the start of this page, on the TI-83 Plus we use the sequence keys to accomplish the same thing.) Figure 2 shows the MATRIX menu. The calculator used here already has a matrix stored in [A], in fact it has a matrix with 10 rows and 7 columns stored in [A]. We will store our matrix in place of that one. To do this, we can press the key to select the [A] matrix name. |
| Figure 3 shows the result of our work in Figure 2. The name [A] has been appended to the store command in our screen. The action has not been performed at this point. We have merely created the command that defines the matrix and assigns it to [A]. To perform that command we press the key. |
| Having performed the command to leave Figure 3 the calculator stores the matrix into [A] and
then displays it on the screen as |
| Our next step is to find the reduced row echelon form of this matrix. We could do this ourselves, using what are known as elementary row operations. However, the TI-83 has a function that will try to produce the reduced row echelon form of the original matrix. That command is rref(). We can find this command in the MATRIX menu. Therefore, we open that menu again, via the key. The result is shown in Figure 5. Note that [A] shows up in the list of matrix names as having 2 rows and 3 columns. |
| Our rref() function is not in the list of names. Rather, we will find it under the MATH option. Therefore, we press the key to move the highlight at the top onto the MATH option, as shown in Figure 6. Now we have a new list of options on the screen. Unfortunately, rref() is not one of them. Therefore, use the key to move the selection highlight down the screen until it the screen appears as shown in Figure 7. |
| In Figure 7 we have located the rref( function. Press to select that option and paste rref( onto the main screen. |
| The full command that we want to construct is rref([A]). In Figure 8 we have the start of this command. Now we need to append the name of our matrix. To do this we will need to open the MATRIX menu again. Press to move to Figure 9. |
| Once again, the list of matrix names is presented. [A] is already highlighted. Therefore, we can press to select [A] and move to Figure 10. |
| Once [A] has been pasted into the command, we press to cpomplete the comamnd, as shown in Figure 10. All that remains is to press to get the calculator to perform the function and to produce the reduced row echelon form of the original matrix [A]. |
| Finally, we have the reduced row echolon form of the original matrix. This new form appears as
0X + 1Y = 5 |
The first 11 Figures on this page demonstated a calculator solution to the given problem. This solution required us to create a matrix to hold the coeeficients and the constants of the two equations. We created that matrix by typing it into the calculator, back in Figure 1, and then storing it into a matrix variable in Figure 2 and 3. There is an alternate method to enter or change a matrix, namely, we can use the matrix editor. Figures 12 through 22 demonstrate using the matrix editor to enter a new matrix into the matrix variable [C]. The new matrix will represent the two equations
| We open the matrix menu by pressing the key. |
| Next we press to move the highlight to the EDIT menu item. In Figure 13 we note that the highlight has moved, and that [A] is selected. We want to select [C]. |
| The key sequence moves the selection highlight to our desired [C]. Press to move to Figure 15. |
| Here the calculator has opened the matrix [C] in the matrix editor. At the moment this is a 1 row and 1 column matrix, and the one element in it is 0. However, the flashing cursor is on the number of rows in the matrix. |
| In Figure 16 we have changed the number of rows in the matrix to 2 by typing and we have started to change the number of columns to 3 by typing . The screen already shows the two rows, but it has not changed to chow 3 columns. We will need to press again to do this and move to Figure 17. |
| Both rows and all three columns are visible in the display. The cursor is on item 1,1 which currently holds 1. We want to have that item be 4. Therefore, we press the key to move to Figure 18. |
| Note that the new value is being formed at the bottom of the screen, but that it has not yet been assigned to the element of the matrix. To signal the end of the number and to have the value placed into the matrix, we press , and move on to Figure 19. |
| Our 4 is in the correct place. THe highlight is on the element in the first row, second column. At the bottom of the screen we see that the current value is 0. We want it to be ^{– }3. We press to enter that value at the bottom of the screen, shown in Figure 20. |
| The ^{– }3 value is shown at the bottom. Press top accept that value and move on to the next matrix element. Thereafter, we continue to enter array elements until we have the values shown in Figure 21. |
| Figure 21 reflects the steps taken to enter the matrix, up to but not inlcuding the entry of the final value, ^{– }50, into the element at row 2 column 3. Once we press to complete the entry of this value, our matrix will be complete. |
| Here we have the completed matrix. Note that the highlight is still on the element in the lst row, last column. We could use the cursor keys to move around within the matrix. Right now we will verify that all of the values in the matrix are correct. |
| Once we are satisfied that hte matrix holds the correct values, we press the sequence to exit the matrix editor. This retuns us to the main screen, which is unchanged since we left after Figure 11. We need to formulate the rref([C]) command. To do this we press to open the matrix menu, shown in Figure 24. |
| We use the key to move the highlight to the MATH menu option. |
| Use over and over until we have the selector highlight on the rref( function. Press to paste that selection onto the main screen, shown in Figure 26. |
| The command has been started. We need to generate the name of our matrix, C, so we press to open the matrix menu again, as shown in Figure 27. |
| We can select the [C] item by pressing the key. That will paste [C] onto the main screen. |
| We complete the command with
and the calculator responds as in Figure 28.
The matrix that we put into [C] represented the two equations
^{– }5x + 7y = ^{– }50 0x + 1y = ^{– }5 |
The main page for solving systems of linear equations on the TI-83 and TI-83 Plus.
The next example page covers a Simple 3 equation 3 variable situation.
©Roger M. Palay
Saline, MI 48176
May, 2001