Simple 3 equation 3 variable, with integer coefficients, fractional answers

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 coefficients, but integer answers.
The next example page covers a Simple 4 equation 4 variable situation, with no 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

and the

key will be shown as

, again to save space.

The problem we will use on this page is 84x + 147y - 105z = -22
42x - 147y + 168z = 94
126x - 84y + 42z = 97

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 3 variables and 4 equations. Therefore we respond with the key to complete Figure 1.
Figure 2 We leave Figure 1 by pressing the key. The calculator shifts to the screen in Figure 2, asking for the coefficients and constant value for the first equation. The equation is 84x + 147y - 105z = -22 which we enter via the keys. This is the state shown in Figure 2.

Figure 3 We leave Figure 2 by pressing the key. The calculator shifts to the screen in Figure 3, asking for the coefficients and constant value for the second equation, namely, 42x - 147y + 168z = 94. The key sequence enters those values and completes Figure 3.

Figure 4 We leave Figure 3 by pressing the key. The calculator shifts to the screen in Figure 4, asking for the third equation, namely, 126x - 84y + 42z = 97. The key sequence enters those values and completes Figure 4.

Figure 5 Pressing the key causes the calculator to produce the answers, shown in Figure 6. In this case, the answers are decimal fractions. The first, .5, is easily recognized as 1/2. The second, -.333333333333 surely represents -1/3. The third value, .142857142857, may not be well known. How do we get the calculator to convert these values into common fractions?

Figure 6 There is nothing that we can do with the display from the SIMULT process, directly. However, the menu items at the bottom of the screen do give us commands to save the values from the problem. "F2" corresponds to a command to "STORE the a's", that is, store the coefficients of the variables into a matrix. "F3" corresponds to a command to "STORE the b's", that is, store the constants of the equations into a vector. And, "F4" corresponds to a command to "STORE the x's", that is, store the answers for the variables into a vector. Once values are stored in such a fashion, then we can use the calculator to convert the values to fractional form. Our concern is with the answers. Therefore, we press the key to store those values. The calculator responds as shown in Figure 7, asking for the name of the variable under which the values are to be saved. It does this by displaying "NAME=" and then waiting for a name, with the calculator in ALPHABETIC mode. In Figure 7 we have given that name as "AA" via the keys .

Figure 7 We can then exit the SIMULT process via the key. This returns us to the screen that we left when we went into the SIMULT process. In the case of Figure 7, that was a clear screen.

Figure 8 We want to convert the vector AA to fractional form. The calculator uses the command to convert values to a fraction. The symbol can be found under the "MATH" "OPS" menu, although you will need to use the "MORE" key to find it in the second sub-menu. However, on the calculator being demonstrated here, the operation has been installed on the "CUSTOM" menu. Therefore, to generate Figure 8, we pressed the keys to produce the "AA". Then we open the CUSTOM menu via the key, displaying the menu seen at the bottom of Figure 8. Finally, pressing the key puts the onto the screen, completing Figure 8.

Figure 9 After prducing the line in Figure 8, press the key to cause the calculator to try to convert the decimals stored in "AA" to common fractions. The result is shown in Figure 9. In this case the answers come out as x=1/2, y=-1/3, and z=1/7.

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 3 variables and 4 equations. Therefore we respond with the key to complete Figure 1.
Figure 2	We leave Figure 1 by pressing the key. The calculator shifts to the screen in Figure 2, asking for the coefficients and constant value for the first equation. The equation is 84x + 147y - 105z = -22 which we enter via the keys. This is the state shown in Figure 2.
Figure 3	We leave Figure 2 by pressing the key. The calculator shifts to the screen in Figure 3, asking for the coefficients and constant value for the second equation, namely, 42x - 147y + 168z = 94. The key sequence enters those values and completes Figure 3.
Figure 4	We leave Figure 3 by pressing the key. The calculator shifts to the screen in Figure 4, asking for the third equation, namely, 126x - 84y + 42z = 97. The key sequence enters those values and completes Figure 4.
Figure 5	Pressing the key causes the calculator to produce the answers, shown in Figure 6. In this case, the answers are decimal fractions. The first, .5, is easily recognized as 1/2. The second, -.333333333333 surely represents -1/3. The third value, .142857142857, may not be well known. How do we get the calculator to convert these values into common fractions?
Figure 6	There is nothing that we can do with the display from the SIMULT process, directly. However, the menu items at the bottom of the screen do give us commands to save the values from the problem. "F2" corresponds to a command to "STORE the a's", that is, store the coefficients of the variables into a matrix. "F3" corresponds to a command to "STORE the b's", that is, store the constants of the equations into a vector. And, "F4" corresponds to a command to "STORE the x's", that is, store the answers for the variables into a vector. Once values are stored in such a fashion, then we can use the calculator to convert the values to fractional form. Our concern is with the answers. Therefore, we press the key to store those values. The calculator responds as shown in Figure 7, asking for the name of the variable under which the values are to be saved. It does this by displaying "NAME=" and then waiting for a name, with the calculator in ALPHABETIC mode. In Figure 7 we have given that name as "AA" via the keys .
Figure 7	We can then exit the SIMULT process via the key. This returns us to the screen that we left when we went into the SIMULT process. In the case of Figure 7, that was a clear screen.
Figure 8	We want to convert the vector AA to fractional form. The calculator uses the command to convert values to a fraction. The symbol can be found under the "MATH" "OPS" menu, although you will need to use the "MORE" key to find it in the second sub-menu. However, on the calculator being demonstrated here, the operation has been installed on the "CUSTOM" menu. Therefore, to generate Figure 8, we pressed the keys to produce the "AA". Then we open the CUSTOM menu via the key, displaying the menu seen at the bottom of Figure 8. Finally, pressing the key puts the onto the screen, completing Figure 8.
Figure 9	After prducing the line in Figure 8, press the key to cause the calculator to try to convert the decimals stored in "AA" to common fractions. The result is shown in Figure 9. In this case the answers come out as x=1/2, y=-1/3, and z=1/7.