TI-83: Simple 3 equation 3 variable, with other variables

The main page for solving systems of linear equations on the TI-83 and TI-83 Plus.
The previous example page covers a Simple 4 equation 4 variable situation.
The next example page covers a Simple 7 equation 7 variable situation.

WARNING: The TI-83 and TI-83 Plus are almost identical in terms of the material presented here. The major difference is the labels that are on certain keys. The TI-83 has a

key, whereas on the TI-83 Plus requires 2 keys to achieve the same result, namely, the

key. The text below will be done from the perspective of the TI-83. That is, all reference to the MATRIX key will be demonstrated via the

key. If the user has a TI-83 Plus then the key strokes should be

. To save some space, and to ignore this difference, the numeric keys (the gray ones) have been changed in some places to show the key face, as in

. In addition, the

key may be shown as

and the

key may be shown as

, again to save space.

The problem we will use on this page is different from those used on earlier pages. Here the problem uses variables other than the usual x, y, and z. In addition, the problem is stated without having the equations in standard form. Thus, for this page, our problem is given as 9m + 5g = 11k - 125
13g + 7k = 10m + 2
5k = 8m + 6g + 60

Our first step is to identify the variables. They are m, g, and k. Our next step is to determine an order for these variables. Any order will do, it does not have to be alphabetic, but whatever order we choose we must maintain for the rest of the problem. We will choose alphabetic simply because it is so easy to remember and verify. Finally, we need to rewrite the equations in standard form, given the order of the variables, g, k, and m, that we have chosen. The new form will be

5g - 11k + 9m = -125
13g + 7k - 10m = 2
-6g + 5k - 8m = 60

We are looking for the values of the variables, g, k, and m, that will make all three equations true. To do this we will create a matrix that represents the equations presented above. In particular, we want the calculator to hold the matrix

5	^–11	9	^–125
13	7	^–10	2
^–6	5	^–8	60

There are two ways to enter a matrix into the calculator. First, you can use the [ and ] characters to type the matrix directly into the calcualtor. That method was demonstrated in the 3 variable, 3 equation page. Second, you can use the Matrix Editor. That is the method presented here, in Figures 1 through 5. The other Figures on this page are used to demonstrate how the calculator produces the reduced row echelon form of the given matrix.

Figure 1 Open the matrix menu with the key. The calculator used here has two previously defined matrices, [A] and [C]. For this problem we will re-use matrix [A].

Figure 2 Use to move the highlight to the EDIT command at the top of the screen. Note that matrix [A] is already selected. Therefore, press to move to Figure 3.

Figure 3 Before we edit the numbers that are in the matix, the calculator allows us to change the size of the matrix. [A] has 3 rows and 4 columns from its earlier definition. We leave these values as they are by pressing .

Figure 4 The calculator matrix editor highlights the number in row 1 column 1 of the matrix. We will enter all 12 of the desired values in order to have the calculator store the matrix

5 ^–11 9 ^–125

13 7 ^–10 2

^–6 5 ^–8 60

Figure 4 shows the rightmost 3 columns of elements in that matrix.

Figure 5 Use the key to move the highlight back to the first column. We verify the contents of [A] and then leave the matrix editor by pressing to return to the main screen, shown in Figure 5.

Figure 6 The calculator used in this example had a blank main screen. In order to try to solve our problem we want to create the command rref([C]). We can find the rref( command in the matrix menu.

Figure 7 Press to open the matrix menu, and to shift the highlight to the MATH menu. Then use the to move down the menu to find the rref( item.

Figure 8 Press to select that item and paste it onto the screen, as shown in Figure 8.

Figure 9 Return to the matrix menu with the key in order to select the name of the matrix, [A] via the key.

Figure 10 Complete the command with the key, and tell the calculator to perform the command with the key. The calculator determines the correct answer and displays it as Figure 10.
We read this display by converting it back to equations
1g + 0k + 0m = ^–9
0g + 1k + 0m = 2
0g + 0k + 1m = ^–3
Therefore, we have a unique solution to all three equations when g = ^–9, k = 2, and m = ^–3.

The main page for solving systems of linear equations on the TI-83 and TI-83 Plus.
The previous example page covers a Simple 4 equation 4 variable situation.
The next example page covers a Simple 7 equation 7 variable situation.