Simple 3 equation 3 variable, parallel planes

The main page for solving systems of linear equations on the TI-85 and TI-86.
The previous example page covers a Simple 2 equation 2 variable situation but with parallel lines.
The next example page covers a Simple 3 equation 3 variable situation, but with no solution since the planes do not intersect, but are not parallel.

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 5x - 7y + 8z = -102
8x + 4y - 7z = 56
10x -14y + 16z = 100

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. The calculator shifts to the screen in Figure 2, asking for the coefficients and constant value for the first equation. The key sequence enters those values and completes Figure 2.

Figure 3 We leave Figure 2 by pressing the key. The calculator shifts to the screen in Figure 3, asking for the coefficients and constant value for the second equation. The key sequence enters those values and completes 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 10, 1-14, 16, and 100. 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 displays the error message in Figure 5.
We could press to quit the SIMULT processing, or we could press to return to the data entry phase of the SIMULT processing, giving us a display that is essentially the same as was given in Figure 2. We would do this so that we could display and change if need be the coefficients and constant terms. In this case, the values were entered correctly. We could simply quit the processing.

If we return to the original problem,

5x - 7y + 8z = -102
8x + 4y - 7z = 56
10x -14y + 16z = 100 we can see that the first and third equations are parallel. The planes that these equations represent will never cross. The left side of the third equation is simply twice the left side of the first equation. However, the right side of the third is not twice the right side of the first. It is true that the second equation intersects the first and the third, but each of those intersections is a straight line that lies in the plane that represents the second equation. In that plane, the two lines are parallel.

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. The calculator shifts to the screen in Figure 2, asking for the coefficients and constant value for the first equation. The key sequence enters those values and completes Figure 2.
Figure 3	We leave Figure 2 by pressing the key. The calculator shifts to the screen in Figure 3, asking for the coefficients and constant value for the second equation. The key sequence enters those values and completes 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 10, 1-14, 16, and 100. 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 displays the error message in Figure 5. We could press to quit the SIMULT processing, or we could press to return to the data entry phase of the SIMULT processing, giving us a display that is essentially the same as was given in Figure 2. We would do this so that we could display and change if need be the coefficients and constant terms. In this case, the values were entered correctly. We could simply quit the processing.