Simple 2 equation 2 variable

The main page for solving systems of linear equations on the TI-85 and TI-86.
The next example page covers a Simple 3 equation 3 variable situation.

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 = 14
5x - 7y = -45 We are looking for the values of the variables that make both equations true. [Remember that these are linear equations in two variables. On the Cartesian plane, each equation represents a straight line. There are an infinite number of points on each line, and an infinite number of solutions to the individual equations. However, these particular lines cross at a point. That point of intersection is a solution to both equations. It is the only point that solves both equations. We need to find that point, that x and y value that solve both equations.]

Before we start using the calculator, note that these equations are given in standard form. That is, they appear as Ax + By = C, where A, B, and C are numeric values. For two equations, in two unknowns (variables) x and y, we could write the equations in a general standard form as:

Ax + By = C
Dx + Ey = F As we will see, the calculator uses a more general standard form for these equations, namely: a_1,1 x₁ + a_1,2 x₂ = b₁
a_2,1 x₁ + a_2,2 x₂ = b₂ This new form can be more confusing in simple cases, such as two variables and two equations, but it is more useful in complex situations, such as 7 variables and 7 equations. The key to understanding this general form is that the numbers after the a's indicate first the equation (row) and second the variable to which the numeric coefficient is attached. Thus, a_2,1 indicates that this is the number in the second equation attached to the first variable. The variables are numbered by the subscript of x, so x₁ represents the first variable and x₂ represents the second variable. The constants on the right side of the equations are numbered by the equation (row) in which they appear. Therefore, b₁ is the constant for the first equation and b₂ is the constant for the second equation.

The problem that we were given was:

3x + 4y = 14
5x - 7y = -45 and we remember that the general standard form on the calculator is: a_1,1 x₁ + a_1,2 x₂ = b₁
a_2,1 x₁ + a_2,2 x₂ = b₂ so, for this problem

a_1,1 is 3	x₁ is x	a_1,2 is 4	x₂ is y	b₁ is 14
a_2,1 is 5	x₁ is x	a_2,2 is -7	x₂ is y	b₂ is -45

With all of that out of the way, we are finally ready to start using the calculator. The steps shown before assume that the calculator is turned on, that we are not in any menu, and that the screen is clear.

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 2 variables and 2 equations.

Figure 2
Since we have 2 variables and 2 equations, we respond with the value 2 by pressing the key. This is shown in Figure 2. Then we conclude our input with the key.

Figure 3 In Figure 2 we told the calculator that we have two variables and two equations. Now, in Figure 3, the calculator is prompting us for the values of the coefficients for those variables, in the first equation. Notice that the calcualtor is using the general standard form for our equations. Thus, the display is asking for the value of a_1,1 which is the coefficient for the first variable in the first equation. That first equation was
3x + 4y = 14 so we will want to respond with the value 3.

Figure 4 Figure 4 shows the display after we have pressed the key followed by the key. Notice that the blinking cursor has moved to the next line, asking for a value for a_1,2 which is the coefficient for the second variable in the first equation.

Figure 5 The values that we want to assign to a_1,2 is 4. To do this we press the key. Then we could press the "enter" button as we did in Figure 4. However, we could also press the key and have the same effect, as shown in Figure 5.
This leaves the calculator asking for the value of b₁, the constant term in the first equation.

Figure 6 Again, our original first equation was
3x + 4y = 14 so we will want to respond with the value 14. The keys produce this value and change the display to Figure 6.
We have entered all of the values for the first equation. Now, if we press either the "enter" or the "down arrow" key, the calculator will display a new screen asking for values for the second equation.

Figure 7 To get from Figure 6 to Figure 7 we pressed the key. The display in Figure 7 is asking for the coefficients and constant for the second equation. Notice that a_2,1 is the coefficient of the first variable in the second equation, a_2,2 is the coefficient of the second variable in the second equation. And, b₂ is the constant value in the second equation. We are ready to assign values to those items.

Figure 8 Our second equation is 5x - 7y = -45 so we want to enter the values 5, -7, and -45. We do that via the keys and . The calculator display should appear as in Figure 8, with the blinking cursor following the -45.
At this point we are done entering all of th evalues for the two equations. We could just select the "SOLVE" option from the menu at this point. However, there is a tendency to press the "ENTER" key to signal the "end of data". Doing so will do no harm and is shown in Figure 9.

Figure 9 The effect of pressing the key at the end of Figure 8 is shown in Figure 9. The blinking cursor returns to the start of the data entry field for the last value. In our case, the cursor is positioned over the negative sign of the -45. This change does not affect the values already entered.

Figure 10 Both Figure 8 and Figure 9 show represent having all of the coefficients and constants entered into the calculator. We are ready to obtain the solution. To do so we need to select the "SOLVE" option from the menu, which we do by pressing the key. The calculator responds with the data in Figure 10, namely that the solution is x₁=-2 and x₂=-5. Remember that x₁ corresponds to the variable x in the original problem. Similarly, x₂ corresponds to the variable y. Therefore, the solution is x=-2 and y=5, or, as an ordered pair, the point (-2,5).

The main page for solving systems of linear equations on the TI-85 and TI-86.
The next example page covers a Simple 3 equation 3 variable situation.

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 2 variables and 2 equations.
Figure 2	Since we have 2 variables and 2 equations, we respond with the value 2 by pressing the key. This is shown in Figure 2. Then we conclude our input with the key.
Figure 3	In Figure 2 we told the calculator that we have two variables and two equations. Now, in Figure 3, the calculator is prompting us for the values of the coefficients for those variables, in the first equation. Notice that the calcualtor is using the general standard form for our equations. Thus, the display is asking for the value of a_1,1 which is the coefficient for the first variable in the first equation. That first equation was 3x + 4y = 14 so we will want to respond with the value 3.
Figure 4	Figure 4 shows the display after we have pressed the key followed by the key. Notice that the blinking cursor has moved to the next line, asking for a value for a_1,2 which is the coefficient for the second variable in the first equation.
Figure 5	The values that we want to assign to a_1,2 is 4. To do this we press the key. Then we could press the "enter" button as we did in Figure 4. However, we could also press the key and have the same effect, as shown in Figure 5. This leaves the calculator asking for the value of b₁, the constant term in the first equation.
Figure 6	Again, our original first equation was 3x + 4y = 14 so we will want to respond with the value 14. The keys produce this value and change the display to Figure 6. We have entered all of the values for the first equation. Now, if we press either the "enter" or the "down arrow" key, the calculator will display a new screen asking for values for the second equation.
Figure 7	To get from Figure 6 to Figure 7 we pressed the key. The display in Figure 7 is asking for the coefficients and constant for the second equation. Notice that a_2,1 is the coefficient of the first variable in the second equation, a_2,2 is the coefficient of the second variable in the second equation. And, b₂ is the constant value in the second equation. We are ready to assign values to those items.
Figure 8	Our second equation is 5x - 7y = -45 so we want to enter the values 5, -7, and -45. We do that via the keys and . The calculator display should appear as in Figure 8, with the blinking cursor following the -45. At this point we are done entering all of th evalues for the two equations. We could just select the "SOLVE" option from the menu at this point. However, there is a tendency to press the "ENTER" key to signal the "end of data". Doing so will do no harm and is shown in Figure 9.
Figure 9	The effect of pressing the key at the end of Figure 8 is shown in Figure 9. The blinking cursor returns to the start of the data entry field for the last value. In our case, the cursor is positioned over the negative sign of the -45. This change does not affect the values already entered.
Figure 10	Both Figure 8 and Figure 9 show represent having all of the coefficients and constants entered into the calculator. We are ready to obtain the solution. To do so we need to select the "SOLVE" option from the menu, which we do by pressing the key. The calculator responds with the data in Figure 10, namely that the solution is x₁=-2 and x₂=-5. Remember that x₁ corresponds to the variable x in the original problem. Similarly, x₂ corresponds to the variable y. Therefore, the solution is x=-2 and y=5, or, as an ordered pair, the point (-2,5).