We are looking for the values of the variables that make all four equations true. [Remember that these are linear equations in four variables. In the earlier example of two variables in two equations, we could associate the problem with lines on in a Cartesian plane where each equation represents a line. In the 3 equation 3 variable situation we could associate each equation with a plane in Cartesian space. For 4 equations and 4 variables we no longer have the luxury of a physical model. Nonetheless the observations of the earlier situations hold true. There are an infinite number of 4 values (one for each of x, y, z, and w) that make each equation true. In the problem given above, there is exactly one set of 4 values that make all four equations true. We need to find that set of 4 values, for x, y, z, and w, that solve all four equations.]
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. |
x_{j} | for the j^{th} variable. Thus, x_{4} is the fourth variable (in our case w). |
b_{i} | is the constant value in the i^{th} equation. Thus b_{3} is the constant value in the third equation. |
Now, onto the problem on the calculator.
| 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. |
| 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. |
| The desired values are 2, 5, -9, 3, and 139. NOTE that there is an error here. We want 151 not 139. However, we are introducing an error now so that we can see the effect and learn how to correct it below. We enter these via the keys . After pressing those keys the screen should appear as in Figure 3. |
| We accept the values of Figure 3 and move to Figure 4 by pressing the key. Notice that the display is now asking for values for the second equation, as indicated by the subscripts. |
| The values for the coefficients and constant in the second equation are 5, 6, -4, 2, and 103. We press and to produce the display shown in Figure 5. |
| We accept the values of Figure 5 and move to Figure 6 by pressing the
key. Notice that the display is now asking for values
for the third equation, as indicated by the subscripts. The values for the coefficients and constant in the third equation are 3, -4, 2, 7, and 16. We press and to produce the display shown in Figure 6. |
| We accept the values of Figure 6 and move to Figure 7 by pressing the
key. Notice that the display is now asking for values
for the fourth equation, as indicated by the subscripts. The values for the coefficients and constant in the fourth equation are 11, 7, 4, -8, and -32. We press and to produce the display shown in Figure 7. |
| After entering all of the data, shown as complete in Figure 7, we
press the key to select the "SOLVE" option from the
menu. The calculator responds with the solution shown in Figure 8.
This is the solution to the equations that we have entered via the coefficients and the constants. Unfortunately, these are the wrong values for our original problem. [This being a problem for a math class, we expect that the answers will be fairly "nice", most likely integer values, or at worst common fractions.] These values do not make the original first equation work. 2*1.0169911505 + 5*8.25663716814 + -4*(-7.70796460177 + 3*8.76991150442 is 138.998... or, essentially, 139. Our original first equation was |
| Let us go back and look at the coefficients that we had entered. Perhaps there was an
error in our data entry. From the menu on Figure 8, we press the
key to select the "COEFS" option. This returns us to the data entry screen for
the first equation, as shown in Figure 9, Note that the blinking cursor is covering the
value 2 assigned to the first coefficient. An examination of this list of coefficients and the constant reveals our error. The value for b_{1} should have been 151, not 139. This is the error introduced back in Figure 3. |
| We can press the "ENTER" key or we can use the "down arrow" key to move to the b_{1}line of data input. Press to change the display to correspond to Figure 10. |
| Now we press the keys for the correct value, , producing the putput in Figure 11. |
| We left Figure 11 with the correct values in place. We press the key to select the "SOLVE" option from the menu. The calculator responds with the solution shown in Figure 12. Thus, there is a solution to all four equations when x_{1} or x is 3, x_{2} or y is 5, x_{3} or z is -11, and x_{4} or w is 7. |
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.
The next example page covers a Simple 3 equation 3 variable situation, with
other variables.
©Roger M. Palay
Saline, MI 48176
October, 1998