Simple 7 equation 7 variable

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 other variables.
The next example page covers a Missing 3 equation 3 variable situation, where some variables may be missing.

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 2s - 7t + 3u + 5v - 7w + 11x + 8y = 114
5s + 7t - u + 9v - 6w + 3x - 5y = -11
13s + 11t + 8u + 7v + 4w + 2x + y = -25
12s + t + 6u + 5v + 9w + 3x + 11y = -37
s - 4t + 7u - 3v + 9w - 2x + 6y = 4
4s + t - 6u - 3v + 8w - 7x - 4y = -91
-7s - 2t + 5u + v + 9w + 3x - 4y = -50

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

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 earliest 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, x₄ is the fourth variable (in our case v).

b_i is the constant value in the i^th equation. Thus b₃ is the constant value in the third equation.

Now, onto the problem on the calculator.

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 7 variables and 7 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. Notice in Figure 2 that the calculator is requesting values for each of the first six coefficients, using the general standard form for our first equation. The remaining coefficient and the constant value do not fit on the screen. The down-arrow to the left of the a1,6 indicates that there are more entries for this screen. Figure 2 also shows that we have supplied the values for the first six coefficients from the first equation 2s - 7t + 3u + 5v - 7w + 11x + 8y = 114. The keys for this are . After pressing those keys the screen should appear as in Figure 2.

Figure 3 We can press the key to move to the next data value. Figure 3 shows the result of pressing that key. The calculator is now ready for the seventh coeffiicient in the first equation.

Figure 4 Figure 4 shows that we have supplied the seventh coefficient via , pressed to move to the constant value, and entered 114 via the keys. This completes the first equation. We can press to move to the next equation.

Figure 5 Figure 5 shows the top part of the screen for the second equation, 5s + 7t - u + 9v - 6w + 3x - 5y = -11 and the values for the first six coefficients have been entered via the keys. We can move to the next coefficient via the key.

Figure 6 Figure 6 completes the second equation via the keys. We can press to move to the next equation.

Figure 7 Figure 7 shows the top part of the screen for the third equation, 13s + 11t + 8u + 7v + 4w + 2x + y = -25 and the values for the first six coefficients have been entered via the keys. We can move to the next coefficient via the key.

Figure 8 Figure 8 completes the third equation via the keys. We can press to move to the next equation.

Figure 9 Figure 9 shows the top part of the screen for the fourth equation, 12s + t + 6u + 5v + 9w + 3x + 11y = -37 and the values for the first six coefficients have been entered via the keys. We can move to the next coefficient via the key.

Figure 10 Figure 10 completes the fourth equation via the keys. We can press to move to the next equation.

Figure 11 Figure 11 shows the top part of the screen for the fifth equation, s - 4t + 7u - 3v + 9w - 2x + 6y = 4 and the values for the first six coefficients have been entered via the keys. We can move to the next coefficient via the key.

Figure 12 Figure 12 completes the fifth equation via the keys. We can press to move to the next equation.

Figure 13 Figure 13 shows the top part of the screen for the sixth equation, 4s + t - 6u - 3v + 8w - 7x - 4y = -91 and the values for the first six coefficients have been entered via the keys. We can move to the next coefficient via the key.

Figure 14 Figure 14 completes the sixth equation via the keys. We can press to move to the last equation.

Figure 15 Figure 15 shows the top part of the screen for the seventh equation, -7s - 2t + 5u + v + 9w + 3x - 4y = -50 and the values for the first six coefficients have been entered via the keys. We can move to the next coefficient via the key.

Figure 16 Figure 6 completes the seventh equation via the keys. We are ready to "solve" the system of linear equations.

Figure 17 After entering all of the data, shown as complete in Figure 16, we press the key to select the "SOLVE" option from the menu. The calculator responds with the solution shown in Figure 17.
This is the solution to the equations that we have entered via the coefficients and the constants. We recall that x₁ is s, x₂ is t, x₃ is u, x₄ is v, x₅ is w, x₆ is x, and x₇ is y. Therefore, the values s=5, t=-7, u=8, v=-6, w=-9, x=2, and y=-3 solve all seven equations.

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, x₄ is the fourth variable (in our case v).
b_i	is the constant value in the i^th equation. Thus b₃ is the constant value in the third equation.

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 7 variables and 7 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. Notice in Figure 2 that the calculator is requesting values for each of the first six coefficients, using the general standard form for our first equation. The remaining coefficient and the constant value do not fit on the screen. The down-arrow to the left of the a1,6 indicates that there are more entries for this screen. Figure 2 also shows that we have supplied the values for the first six coefficients from the first equation 2s - 7t + 3u + 5v - 7w + 11x + 8y = 114. The keys for this are . After pressing those keys the screen should appear as in Figure 2.
Figure 3	We can press the key to move to the next data value. Figure 3 shows the result of pressing that key. The calculator is now ready for the seventh coeffiicient in the first equation.
Figure 4	Figure 4 shows that we have supplied the seventh coefficient via , pressed to move to the constant value, and entered 114 via the keys. This completes the first equation. We can press to move to the next equation.
Figure 5	Figure 5 shows the top part of the screen for the second equation, 5s + 7t - u + 9v - 6w + 3x - 5y = -11 and the values for the first six coefficients have been entered via the keys. We can move to the next coefficient via the key.
Figure 6	Figure 6 completes the second equation via the keys. We can press to move to the next equation.
Figure 7	Figure 7 shows the top part of the screen for the third equation, 13s + 11t + 8u + 7v + 4w + 2x + y = -25 and the values for the first six coefficients have been entered via the keys. We can move to the next coefficient via the key.
Figure 8	Figure 8 completes the third equation via the keys. We can press to move to the next equation.
Figure 9	Figure 9 shows the top part of the screen for the fourth equation, 12s + t + 6u + 5v + 9w + 3x + 11y = -37 and the values for the first six coefficients have been entered via the keys. We can move to the next coefficient via the key.
Figure 10	Figure 10 completes the fourth equation via the keys. We can press to move to the next equation.
Figure 11	Figure 11 shows the top part of the screen for the fifth equation, s - 4t + 7u - 3v + 9w - 2x + 6y = 4 and the values for the first six coefficients have been entered via the keys. We can move to the next coefficient via the key.
Figure 12	Figure 12 completes the fifth equation via the keys. We can press to move to the next equation.
Figure 13	Figure 13 shows the top part of the screen for the sixth equation, 4s + t - 6u - 3v + 8w - 7x - 4y = -91 and the values for the first six coefficients have been entered via the keys. We can move to the next coefficient via the key.
Figure 14	Figure 14 completes the sixth equation via the keys. We can press to move to the last equation.
Figure 15	Figure 15 shows the top part of the screen for the seventh equation, -7s - 2t + 5u + v + 9w + 3x - 4y = -50 and the values for the first six coefficients have been entered via the keys. We can move to the next coefficient via the key.
Figure 16	Figure 6 completes the seventh equation via the keys. We are ready to "solve" the system of linear equations.
Figure 17	After entering all of the data, shown as complete in Figure 16, we press the key to select the "SOLVE" option from the menu. The calculator responds with the solution shown in Figure 17. This is the solution to the equations that we have entered via the coefficients and the constants. We recall that x₁ is s, x₂ is t, x₃ is u, x₄ is v, x₅ is w, x₆ is x, and x₇ is y. Therefore, the values s=5, t=-7, u=8, v=-6, w=-9, x=2, and y=-3 solve all seven equations.