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
ai,j | for the coefficient of the jth variable in the ith equation. Thus, a2,3 is the coefficient for the 3rd variable in the 2nd equation. |
xj | for the jth variable. Thus, x4 is the fourth variable (in our case w). |
bi | is the constant value in the ith equation. Thus b3 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 ![]() ![]() ![]() ![]() ![]() |
![]() | We leave Figure 1 by pressing the ![]() |
![]() | 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
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() | We accept the values of Figure 3 and move to Figure 4 by pressing the
![]() |
![]() | The values for the coefficients and constant in the second equation are
5, 6, -4, 2, and 103. We press
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() | We accept the values of Figure 5 and move to Figure 6 by pressing the
![]()
The values for the coefficients and constant in the third equation are
3, -4, 2, 7, and 16. We press
|
![]() | We accept the values of Figure 6 and move to Figure 7 by pressing the
![]()
The values for the coefficients and constant in the fourth equation are
11, 7, 4, -8, and -32. We press
|
![]() | After entering all of the data, shown as complete in Figure 7, we
press the ![]() 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 ![]() An examination of this list of coefficients and the constant reveals our error. The value for b1 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
b1line of data input. Press
![]() ![]() ![]() ![]() |
![]() | Now we press the keys for the correct value,
![]() ![]() ![]() |
![]() | We left Figure 11 with the correct values in place. We
press the ![]() |
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