Simple 4 equation 4 variable, but no solution.

Simple 4 equation 4 variable, but no solution

The main page for solving systems of linear equations on the TI-85 and TI-86.
The previous example page covers a Simple 3 equation 3 variable situation, with fractional answers.
The next example page covers a Complex 3 equation 2 variable situation, wihere there is no unique solution.

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 3x + 5y + 2z - 7w = 10
4x - 2y + 11z - 3w = 71
5x - 6y - 3z - 5w = 43
x + 16y + 7z - 9w = -23

We are looking for the values of the variables that make all four equations true.

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 4 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. In that figure we have also entered the desired values, 3, 5, 2, 7, and 10, via the and keys.

Figure 3 We leave Figure 2 by pressing the key. That will cause the display to change to Figure 3. In that figure we have also entered the desired values, 4, -2, 11, -3, and 71, via the and keys. After pressing those keys the screen should appear as in Figure 3.

Figure 4 We accept the values of Figure 3 and move to Figure 4 by pressing the key. That will cause the display to change to Figure 4. In that figure we have also entered the desired values, 5, -6, -3, -5, and 43, via the and keys. After pressing those keys the screen should appear as in Figure 4.

Figure 5 The values for the coefficients and constant in the fourth equation are 1, 16, 7, -9, and 23. We press and to produce the display shown in Figure 5. At this point we are ready to ask the calculator to solve the problem.

Figure 6 We press the key, and the calculator responds with the error message in Figure 6. This message means that there is no unique solution. There may be an infinite set of values that work, or there may be no values that work. In either case, we do not have a unique soltuion.

If we go back tot he original problem, we may be able to see why there is no unique solution. The original statement of the equations was:

3x + 5y + 2z - 7w = 10
4x - 2y + 11z - 3w = 71
5x - 6y - 3z - 5w = 43
x + 16y + 7z - 9w = -23 As we examine this set of equations we might notice that the fourth equation is the difference between twice the first equation and the third equation. In such a situation, it is most likely that there are an infinite number of solutions. It may take a while to find them, but they do exist. For example, if we assume that w=-3, then wecan re-write the first three equations as 3x + 5y + 2z - 7(-3) = 10
4x - 2y + 11z - 3(-3) = 71
5x - 6y - 3z - 5(-3) = 43 which simplifies to 3x + 5y + 2z + 21 = 10
4x - 2y + 11z + 9 = 71
5x - 6y - 3z + 15 = 43 which can be re-written as 3x + 5y + 2z = -11
4x - 2y + 11z = 62
5x - 6y - 3z = 28 These three equations have a solution at x=2, y=-5, and z=4. Therefore, the original equations have a solution at x=2, y=-5, z=4, and w=-3. In a similar fashion we can find that if w=-4 then we have a solution at x=238/523, y=-2902/523, z=2191/523, and w=-4 or if w=-5 we have a solution at x=-5708/523, y=--3189/523, z=2290/523, and w=-5 or, finally, if w=520 then there is a solution at x=810, y=282, z=-95, and w=520. Clearly, the calculator and the SIMULT function are invaluable in finding these points.

The main page for solving systems of linear equations on the TI-85 and TI-86.
The previous example page covers a Simple 3 equation 3 variable situation, with fractiomal answers.
The next example page covers a Complex 3 equation 2 variable situation, wihere there is no unique solution.

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 4 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. In that figure we have also entered the desired values, 3, 5, 2, 7, and 10, via the and keys.
Figure 3	We leave Figure 2 by pressing the key. That will cause the display to change to Figure 3. In that figure we have also entered the desired values, 4, -2, 11, -3, and 71, via the and keys. After pressing those keys the screen should appear as in Figure 3.
Figure 4	We accept the values of Figure 3 and move to Figure 4 by pressing the key. That will cause the display to change to Figure 4. In that figure we have also entered the desired values, 5, -6, -3, -5, and 43, via the and keys. After pressing those keys the screen should appear as in Figure 4.
Figure 5	The values for the coefficients and constant in the fourth equation are 1, 16, 7, -9, and 23. We press and to produce the display shown in Figure 5. At this point we are ready to ask the calculator to solve the problem.
Figure 6	We press the key, and the calculator responds with the error message in Figure 6. This message means that there is no unique solution. There may be an infinite set of values that work, or there may be no values that work. In either case, we do not have a unique soltuion.