SLOPE program for the TI-86 (85)

We are often given the problem of finding the equation of a line given two distinct points on the line. The solution is almost a mechanical process. The following flowchart illustrates the steps:

Such an algorithm can be programmed into the calculator. The SLOPE program on the TI-86 (or on the TI-85) implements such an algorithm. Note that the SLOPE program uses the TOSTR program in order to generate nice looking output. The TI-86 version of the SLOPE program is available as slope.86p and the TI-85 version of the SLOPE program is available as slope.85p. The versions of TOSTR, presented in the 208604.htm page, are available as tostr.86p and tostr.85p.

Figure 1 We can start by pressing the key to open the PRGM menu. Then we press the key to open the NAMES sub-menu. The calculator used to generate Figure 1 holds a large number of programs. The program names (or abbreviations of those names) are displayed in alphabetic order. We will use the key to shift the sub-menu display untilt he SLOPE program is visible.

Figure 2 Figure 2 reflects the condition of having found the SLOPE program in the sub-menu, and of pressing the key to select that program and to paste its name onto the screen.
We can execute the command and start running the program by pressing the key.

Figure 3 In Figure 3, the program has started. The program clears the screen and displays "Get Slope, etc." on the top line, followed by a prompt for the value of "x1" on the second line. We are looking for the solution to "Find the equation of the line containing the points (4,7) and (10,1)." Therefore, we respond with for x1. The program accepts that value and continues by asking for the value for y1. assigns 7 to y1. This is followed by to give x2 the value 10, and to leave the screen as it appears in Figure 3.

Figure 4 We moved from Figure 3 to Figure 4 by pressing the key, and then waiting for the calculator to do its work. In response the calculator computes the slope, given as ^–1 in Figure 4, and it then continues to generate the slope-intercept and the standard form of the resulting equation.
At that point the program is in a "paused" condition, waiting for us to press the ENTER key to continue. The "moving" line of dots in the upper right corner of the screen indicate that "paused" condition.
We press to move to Figure 5.

Figure 5 The program has started over, from the beginning. The screen has been cleared, the heading has been rewritten, and the program prompts for values of the coordinates of our two points. In this case we are solving the problem: "Find the equation of the line containing the points (5,1) and (8,10)." Once the values have been given, we press to accept the final value and move to Figure 6.

Figure 6 The calculator has determined the slope, the slope-intercept form of the solution, and the standard form of the solution.
Please note that the SLOPE program produces a standard form version of the equation that does not require the leading coefficient to be non-negative.

Figure 7 Figure 7 represents the data input for yet another example: "Find the equation of the line containing the points (2,7) and (5,9)."

Figure 8 The calculator has determined the slope between the two points to be 2/3. Note that the calculator program has taken the liberty of writng the slope-intercept form as y=2/3 x + 17/3. It would have been better to have included parentheses so that the answer appears as y=(2/3) x + 17/3, leaving no doubt but that the "x" is not in the denominator of the fraction. Once we see and understand the somewhat "sloppy" form that the program uses for fractional slopes, we can correctly understand the equations that the program produces.

Figure 9 Figure 9 represents the data input for yet another example: "Find the equation of the line containing the points (^–4,3) and (^–7,25)."

Figure 10 The calculator has determined the slope between the two points to be ^–22/3. Note that in the slope-intercept form of the solution equation, the calculator program has left the slope and the second coordinate of the y-intercept as improper fractions. SLOPE does not convert these to either mixed numbers or to decimal values.

Figure 11 Figure 11 represents the data input for yet another example: "Find the equation of the line containing the points (2,9) and (8,9)." By inspection we can see that the solution will be a horizontal line. Let us see how the SLOPE program responds to these values.

Figure 12 The SLOPE program correctly identifies the equation as the constant function, with the solution equation being y=9. Note that we could write this as y=0x+9 to produce the strict slope intercept form. In addition we could write this as 0x + 1y = 9 to produce the strict standard form of the equation.

Figure 13 Figure 13 represents the data input for yet another example: "Find the equation of the line containing the points (6,1) and (6,10)." By inspection we can see that the solution will be a vertical line. Let us see how the SLOPE program responds to these values.

Figure 14 The SLOPE program correctly identifies the equation as a vertical line with the solution equation being x=6. Note that we could write this as 1x + 0y = 6 to produce the strict standard form of the equation. It is impossible to construct a slope-intecept form of this equation: the slope is undefined and the coefficient of y is 0.

Figure 15 If we want to stop the program we need to break out of it. We can press the key to break out of the program. The result is shown in Figure 15. There are only two options. Either we can GOTO or QUIT. We want to QUIT, so we press the key to move to Figure 16.

Figure 16 Figure 16 looks remarkably similar to Figure 14. However, in Figure 14 the calculator was in a "paused" condition. In Figure 16 we are out of the program and ready to do anything that we want to do.

The SLOPE program is designed to generate an equation if we are given the coordinates of two points on the line. How can we use the SLOPE program to solve problems such as

Find the equation of the line that has slope=4/7 and that contains the point (9,6).

In order to use the SLOPE program, we need to know two points on the line. The problem statement gives us one point, (9,6). Can we get a second point? Yes. We know the slope and we know that

m =	4	=	change in y

	7		change in x

Therefore, if we start at the point (9,6) and we add 7 to the x value and add 4 to the y value we produce a new point (16,10) that must be on the desired line. Thus, we use the definition of slope to produce a second point, and this allows us to use the SLOPE program.

Figure 17 Figure 17 shows that the program has been restarted. Having broken out of the program in Figures 15 and 16, we only needed to press the key to re-issue the the last command, namely SLOPE, and therefore restart the program. In Figure 17 we have entered the two points derived from the discussion above. We can press to accept the value for y₂ and to move to Figure 18.

Figure 18 The first thing to notice in Figure 18 is that the slope has been computed to be exactly the value given in the problem statement. This is verification that we have correctly computed the coordinates of the second point. Then we can look at the slope-intercept and standard forms for the desired equation.

Figure 19 The next problem to consider is
Find the equation of the line that has slope=2 and that contains the point (^–4,1).
In order to use the SLOPE program, we need to know two points on the line. The problem statement gives us one point, (^–4,1). Can we get a second point? Yes. We know the slope and we know that
m = 2 = change in y

1 change in x
Therefore, if we start at the point (^–4,1) and we add 1 to the x value and add 2 to the y value we produce a new point (^–3,3) that must be on the desired line. Thus, we use the definition of slope to produce a second point, and this allows us to use the SLOPE program. The values for the two points have been entered into the program in Figure 19.

Figure 20 Figure 20 confirms the second point by calculating the expected slope. Then, the SLOPE program produces the slope-intercept and the standard forms of the desired equations.

Figure 21 The next problem to consider is
Find the equation of the line that has slope=^–3/5 and that contains the point (^–6,^–13).
In order to use the SLOPE program, we need to know two points on the line. The problem statement gives us one point, (^–6,^–13). Can we get a second point? Yes. We know the slope and we know that
m = ^–3 = change in y

5 change in x
Therefore, if we start at the point (^–6,^–13) and we add 5 to the x value and add ^–3 to the y value we produce a new point (^–1,^–16) that must be on the desired line. Thus, we use the definition of slope to produce a second point, and this allows us to use the SLOPE program. The values for the two points have been entered into the program in Figure 21.

Figure 22 Figure 20 confirms the second point by calculating the expected slope. Then, the SLOPE program produces the slope-intercept and the standard forms of the desired equations.

Figure 1	We can start by pressing the key to open the PRGM menu. Then we press the key to open the NAMES sub-menu. The calculator used to generate Figure 1 holds a large number of programs. The program names (or abbreviations of those names) are displayed in alphabetic order. We will use the key to shift the sub-menu display untilt he SLOPE program is visible.
Figure 2	Figure 2 reflects the condition of having found the SLOPE program in the sub-menu, and of pressing the key to select that program and to paste its name onto the screen. We can execute the command and start running the program by pressing the key.
Figure 3	In Figure 3, the program has started. The program clears the screen and displays "Get Slope, etc." on the top line, followed by a prompt for the value of "x1" on the second line. We are looking for the solution to "Find the equation of the line containing the points (4,7) and (10,1)." Therefore, we respond with for x1. The program accepts that value and continues by asking for the value for y1. assigns 7 to y1. This is followed by to give x2 the value 10, and to leave the screen as it appears in Figure 3.
Figure 4	We moved from Figure 3 to Figure 4 by pressing the key, and then waiting for the calculator to do its work. In response the calculator computes the slope, given as ^–1 in Figure 4, and it then continues to generate the slope-intercept and the standard form of the resulting equation. At that point the program is in a "paused" condition, waiting for us to press the ENTER key to continue. The "moving" line of dots in the upper right corner of the screen indicate that "paused" condition. We press to move to Figure 5.
Figure 5	The program has started over, from the beginning. The screen has been cleared, the heading has been rewritten, and the program prompts for values of the coordinates of our two points. In this case we are solving the problem: "Find the equation of the line containing the points (5,1) and (8,10)." Once the values have been given, we press to accept the final value and move to Figure 6.
Figure 6	The calculator has determined the slope, the slope-intercept form of the solution, and the standard form of the solution. Please note that the SLOPE program produces a standard form version of the equation that does not require the leading coefficient to be non-negative.
Figure 7	Figure 7 represents the data input for yet another example: "Find the equation of the line containing the points (2,7) and (5,9)."
Figure 8	The calculator has determined the slope between the two points to be 2/3. Note that the calculator program has taken the liberty of writng the slope-intercept form as y=2/3 x + 17/3. It would have been better to have included parentheses so that the answer appears as y=(2/3) x + 17/3, leaving no doubt but that the "x" is not in the denominator of the fraction. Once we see and understand the somewhat "sloppy" form that the program uses for fractional slopes, we can correctly understand the equations that the program produces.
Figure 9	Figure 9 represents the data input for yet another example: "Find the equation of the line containing the points (^–4,3) and (^–7,25)."
Figure 10	The calculator has determined the slope between the two points to be ^–22/3. Note that in the slope-intercept form of the solution equation, the calculator program has left the slope and the second coordinate of the y-intercept as improper fractions. SLOPE does not convert these to either mixed numbers or to decimal values.
Figure 11	Figure 11 represents the data input for yet another example: "Find the equation of the line containing the points (2,9) and (8,9)." By inspection we can see that the solution will be a horizontal line. Let us see how the SLOPE program responds to these values.
Figure 12	The SLOPE program correctly identifies the equation as the constant function, with the solution equation being y=9. Note that we could write this as y=0x+9 to produce the strict slope intercept form. In addition we could write this as 0x + 1y = 9 to produce the strict standard form of the equation.
Figure 13	Figure 13 represents the data input for yet another example: "Find the equation of the line containing the points (6,1) and (6,10)." By inspection we can see that the solution will be a vertical line. Let us see how the SLOPE program responds to these values.
Figure 14	The SLOPE program correctly identifies the equation as a vertical line with the solution equation being x=6. Note that we could write this as 1x + 0y = 6 to produce the strict standard form of the equation. It is impossible to construct a slope-intecept form of this equation: the slope is undefined and the coefficient of y is 0.
Figure 15	If we want to stop the program we need to break out of it. We can press the key to break out of the program. The result is shown in Figure 15. There are only two options. Either we can GOTO or QUIT. We want to QUIT, so we press the key to move to Figure 16.
Figure 16	Figure 16 looks remarkably similar to Figure 14. However, in Figure 14 the calculator was in a "paused" condition. In Figure 16 we are out of the program and ready to do anything that we want to do.