Simple 3 equation 3 variable, with other variables

The main page for solving systems of linear equations on the TI-85 and TI-86.
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-85 and TI-86 are almost identical in their use of the SIMULT function. The major difference is the labels that are on certain keys. On the TI-85, SIMULT is the 2nd function on the

key, whereas on the TI-86 SIMULT is the 2nd function on the

key. When a difference is important it will be presented in the text below. The exception to this is the "3" key. On the TI-85 it appears as

, while on the TI-86 it is

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

. In addition, the

key will 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.

Before we actually start using the calculator, remember that the calculator will be using a general form for each of the equations, expecting the equations to have the variables is the same order. The earlier pages had much longer explanations of this. Here we will just point out that the calculator will use

a_i,j for the coefficient of the j^th variable in the i^th equation. Thus, a_2,3 is the coefficient for the 3^rd variable in the 2^nd equation. For this problem, that value is -10.

x_j for the j^th variable. Thus, x₂ is the second variable (in our case k).

b_i is the constant value in the i^th equation. Thus b₃ is the constant value in the third equation. In this case, that value is 60.

Now, onto the problem on the calculator.

Figure 1 The keystrokes to start this process are the same on the two calculators, although the keys have a different name. For the TI-85 we start with and , but for the TI-86 we start with and . On either calculator this selects the "SIMULT" function. The calculator responds with a request for the value of "Number" as shown in Figure 1. The SIMULT function expects to have exactly the same number of equations as we have variables. For our problem, we have 3 variables and 4 equations. Therefore we respond with the key to complete Figure 1.
Figure 2 We leave Figure 1 by pressing the key. That will cause the display to change to Figure 2. Notice in Figure 2 that the calculator is requesting values for each of the coefficients and constants that we have in the general standard form for our first equation. The first subscript on each of the "a's" and the subscript on the "b" indicates that we are looking at values for the first equation.
Remember that we need to put the values in according to the standard form. Therefore we want the values 5, -11, 9, and -125. The key sequence accomplishes this and leaves the display as in Figure 2.

Figure 3 We move from Figure 2 to Figure 3 by first pressing the key. Here we need to enter the coefficients and constants for the second equation, in our standard form, namely 13, 7, -10, and 2. We use the keys to complete the image of Figure 3.
Now we can move to the next screen by pressing the key.

Figure 4 In Figure 4 we need to enter the coefficients and constants for the third equation, in our standard form, namely -6, 5, -8, and 60. We use the keys to complete the image of Figure 4.
We have entered all of the values. We are ready to solve the system of linear equations.

Figure 5 We request a solution by pressing the key. The calculator determines the correct answer and displays it as Figure 5.
Once again we need to return to our standard form and recognize that x₁ is g, x₂ is k, and x₃ is m. Therefore, we have a unique solution to all three equations when g=-9, k=2, and m=-3.

a_i,j	for the coefficient of the j^th variable in the i^th equation. Thus, a_2,3 is the coefficient for the 3^rd variable in the 2^nd equation. For this problem, that value is -10.
x_j	for the j^th variable. Thus, x₂ is the second variable (in our case k).
b_i	is the constant value in the i^th equation. Thus b₃ is the constant value in the third equation. In this case, that value is 60.

Figure 1	The keystrokes to start this process are the same on the two calculators, although the keys have a different name. For the TI-85 we start with and , but for the TI-86 we start with and . On either calculator this selects the "SIMULT" function. The calculator responds with a request for the value of "Number" as shown in Figure 1. The SIMULT function expects to have exactly the same number of equations as we have variables. For our problem, we have 3 variables and 4 equations. Therefore we respond with the key to complete Figure 1.
Figure 2	We leave Figure 1 by pressing the key. That will cause the display to change to Figure 2. Notice in Figure 2 that the calculator is requesting values for each of the coefficients and constants that we have in the general standard form for our first equation. The first subscript on each of the "a's" and the subscript on the "b" indicates that we are looking at values for the first equation. Remember that we need to put the values in according to the standard form. Therefore we want the values 5, -11, 9, and -125. The key sequence accomplishes this and leaves the display as in Figure 2.
Figure 3	We move from Figure 2 to Figure 3 by first pressing the key. Here we need to enter the coefficients and constants for the second equation, in our standard form, namely 13, 7, -10, and 2. We use the keys to complete the image of Figure 3. Now we can move to the next screen by pressing the key.
Figure 4	In Figure 4 we need to enter the coefficients and constants for the third equation, in our standard form, namely -6, 5, -8, and 60. We use the keys to complete the image of Figure 4. We have entered all of the values. We are ready to solve the system of linear equations.
Figure 5	We request a solution by pressing the key. The calculator determines the correct answer and displays it as Figure 5. Once again we need to return to our standard form and recognize that x₁ is g, x₂ is k, and x₃ is m. Therefore, we have a unique solution to all three equations when g=-9, k=2, and m=-3.