Complex 4 equations in 3 variables situation, and there is a solution

The main page for solving systems of linear equations on the TI-85 and TI-86.
The previous example page covers a Complex 3 equation 2 variable situation, where two lines are identical and there is a solution.
The next example page covers a Complex 4 equation 3 variable situation, without a 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 + 4y - z = 1
5x - 2y + 7z = 19
2x - y - z = -12
7x + 3y + 2z = 5 Similar to the previous examples, we have more equations than we have variables. In this example there are three variables, but we have 4 equations. The "SIMULT" function always expects the same number of equations as we have variables. In a real sense, we have an extra equation here. If we only had three equations then we could use "SIMULT" to find their point of intersection. Let us move ahead in that direction. At first we will restrict our analysis to the first three equations: 3x + 4y - z = 1
5x - 2y + 7z = 19
2x - y - z = -12

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 restricted version of the problem, we have 3 variables and 3 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, 4, -1, and 1, 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, 5, -2, 7, and 19, via the and keys. After pressing those keys the screen should appear as in Figure 3.

Figure 4 We leave Figure 3 by pressing the key. That will cause the display to change to Figure 4. In that figure we have also entered the desired values, 2, -1, -1, and -12, via the and keys. After pressing those keys the screen should appear as in Figure 4.

Figure 5 At this point we are ready to ask the calculator to solve the problem. We press the key, and the calculator responds with the solution as shown in Figure 5. The solution, x₁=-2, x₂=3, and x₃=5, translates, in the restricted version of the equations as x=-2, y=3, and z=5. This is the point of intersection for the first three equations.

It is nice to have a solution to the first three equations. Having that intersection means that the point (-2,3,5) solves all three of these equations. If we were to graph the first three equations, each equation represents a plane in space. The three planes would cross at the point (-2,3,5). What about the fourth equation? If we were to graph the fourth equation,

7x + 3y + 2z = 5 would it cross the other three at the same point? One way to answer this is to try the values (-2,3,5) in the fourth equation. If it works, then (-2,3,5) will solve all four equations and it would be the solution to the problem. In fact, 7(-2)+3(3)+2(5) is 5. Therefore, (-2,3,5) does satisfy the fourth equation, and it is the unique solution to the system of fourth equations 3x + 4y - z = 1
5x - 2y + 7z = 19
2x - y - z = -12
7x + 3y + 2z = 5 .

If the pattern in the previous pages is to be maintained, then another way to see (-2,3,5) as the solution is to re-use the "SIMULT" operation to see where the fourth equation intersects with two of the first three equations. We can replace the coefficients and constant of the first equation by those of the fourth equation. The keystrokes and screens needed to do this are given below.

Figure 6 We had the solution to the first three equations displayed in Figure 5. Now we want to return to the data entry screen. To do this we select the "COEFS" command from the menu by pressing the key. The result is shown in Figure 6. The calculator has returned to the screen where we enter the coefficients and constant for the first equation. The blinking cursor is covering the 3.

Figure 7 Now we need to enter the coefficients and the constant for the third equation, namely, 7, 3, 2, and 5. We do this via the keys and . The result is shown in Figure 7.

Figure 8 Figure 7 shows that we have entered the values of the fourth equation. The coefficients for the second and third equations are already in the machine. Therefore, we need only select the "SOLVE" command from the menu, via the key, to have the calculator find the intersection of the fourth and the second and third equations. Figure 8 shows that those three equations have the point (-2,3,5) in common. Since this is the point where the first, second, and third equations cross, we can see that the four equations do have a single point in common.

At this point we have shown in two ways that the three original equations do have a single point in common, that they do have a unique solution. For completeness, the presentation could continue with the computation of the intersection of other combinations of the three equations from the original four. For example, we could look at the intersection of the first, second, and fourth equations. This would be a good check to be sure that we had entered the coefficients and constants correctly above. However, we will not do this here.

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 restricted version of the problem, we have 3 variables and 3 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, 4, -1, and 1, 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, 5, -2, 7, and 19, via the and keys. After pressing those keys the screen should appear as in Figure 3.
Figure 4	We leave Figure 3 by pressing the key. That will cause the display to change to Figure 4. In that figure we have also entered the desired values, 2, -1, -1, and -12, via the and keys. After pressing those keys the screen should appear as in Figure 4.
Figure 5	At this point we are ready to ask the calculator to solve the problem. We press the key, and the calculator responds with the solution as shown in Figure 5. The solution, x₁=-2, x₂=3, and x₃=5, translates, in the restricted version of the equations as x=-2, y=3, and z=5. This is the point of intersection for the first three equations.

Figure 6	We had the solution to the first three equations displayed in Figure 5. Now we want to return to the data entry screen. To do this we select the "COEFS" command from the menu by pressing the key. The result is shown in Figure 6. The calculator has returned to the screen where we enter the coefficients and constant for the first equation. The blinking cursor is covering the 3.
Figure 7	Now we need to enter the coefficients and the constant for the third equation, namely, 7, 3, 2, and 5. We do this via the keys and . The result is shown in Figure 7.
Figure 8	Figure 7 shows that we have entered the values of the fourth equation. The coefficients for the second and third equations are already in the machine. Therefore, we need only select the "SOLVE" command from the menu, via the key, to have the calculator find the intersection of the fourth and the second and third equations. Figure 8 shows that those three equations have the point (-2,3,5) in common. Since this is the point where the first, second, and third equations cross, we can see that the four equations do have a single point in common.