Solve By Graphing

Return to Roadmap 2

Start by solving by graphing Example 2 from the Chapter 2 Roadmap page. Start with the initial system of two linear equations in two variables.

1x – 4y	=	20	(1)
3x + 4y	=	12	(2)

Well, actually, we will use the calculator to help in doing this. The TI-84 Plus C has better resolution and color, but for this page we will use a TI-84 Plus.

To get the equations into the calculator we need to solve them for y. Thus, our two equations become:

y	=	(1x – 20) / 4	(3)
y	=	(^–3x+12) / 4	(4)

Figure 1	We use the keys to open the "Y=" screen shown in Figure 1. In this case, no previous functions have been defined.
Figure 2	Then enter our equations, (3) and (4), into the calculator, as shown in Figure 2.
Figure 3	Press the key to display the choices shown in Figure 3. Of those choices we want the Standard option. To get that we just need to press the key. This will open the graph window and the calculator will graph the two defined functions. This is shown in Figure 4.
Figure 4	We can get a good idea of the point of intersection from this graph. It does appear to be at or really near the point (8,^–3). Let us repeat the image shoen in Figure 4, but this time with horizontal and vertical lines added to the image.
Figure 4 doctored up	In this graph we get a slightly better view of the solution. It is clear, however, that we do not need most of the graph area. The Standard window is nice, but we really just need to look at Quadrant IV.
Figure 5	Press the key to change the window settings. Figure 5 shows the settings assoociated with the Standard zoom setting.
Figure 6	Figure 6 shows some new settings so that we are looking mostly at Quadrant IV. Once those values have been entered, press to draw a new graph with the new settings. This is shown in Figure 7.
Figure 7	Now it is even more clear that we have the two lines intersecting at the point (8,^–3).

Figure 7a	Figure 7a is just a doctored version of Figure 7 so that we can more easily verify the values for the point of intersection. If we know that thesolution is at an integer point then it is easy to see that this must be (8,^–3). However, if the real solution had been at (8.0423,^–3.00457) then finding those values would have been impossible via nspecting the graph. Fortunately the calculator has some other features to help us.
Figure 8	We initialie the TRACE feature by pressing . This results in the small changes noted in Figure 8. The calculator is tracing the first equation, the one shown at the top of the screen. The current highlighted point is at X=4.5 and y=^–3.875 as shown at the bottom of the window.
Figure 9	We can repeatedly press the key to shift the highlighted point to the right. After a number of such presses we arrive at Figure 9
Figure 10	If we keep moving the highlighted point to the right we can get just about on top of the point of intersection, as shown in Figure 10. There we can see that the actual highlighted point is at x=8.0106383 and y=^–2.99734. The reason that we "missed" the point of intersection is that the highlighted point moves in small steps so that it is reporting the x and y value at the center of the pixel being highlighted. The actual point of intersection is not at the center of any such pixel.
Figure 11	While we are in TRACE mode, we do have an additional feature on the calculator that makes it easy for us to evaluate the y value on the line that is associated with any specific x value that we choose. To do this we just enter the desired x value. Figure 11 shows us entering the value 8. Once we have done that we just press the key to move to Figure 12.
Figure 12	In moving to Figure 12 the calculator first determines the y value associated with the x value of 8, according to the equation being traced. For us, in this case, that y value is ^–3, and it is displayed at the bottom of the screen. Then the calculator positions the highlight on the pixel that that contains the point (8,^–3).
Figure 13	In preparation for the next pat of the discussion of TRACE mode, we have used the key many times to shift the highlighted point to the left along the specified equation.
Figure 14	Now, if we press the key the highlight changes to the other equation. Note that the second equation is displayed at the top of the screen.
Figure 15	Up to this point we have been using the TRACE feature to explore values along the two functions that we have defined and graphed. Now we will use another feature, one of the features on the CALC menu. To get to the CALC menu, shown in Figure 15, press . From this menu we want to select fifth item, the intersect option. To do that we just need to press the key.
Figure 16	Doing so takes us back to the graph, but now the calculator is asking us to move the highlight onto the First curve. We can use the up and down arrow keys to move the highlight to a different curve if we want to. [Since we have only defined two equations it seems kind of silly for the calculator to ask us to identify which cureves we want to use. But such is the working of the calculator.] We can just press the key to select the line we are currently on.
Figure 17	This takes us to Figure 17 where we see that the first curve has been left with a make on it and the highlight has moved to the second curve. The calculator wants us to identify which is the Second curve. Since the highlight is now on that second curve we just press the key to move ahead in the process.
Figure 18	Figure 18 shows both curves marked and the highlight remains on the second curve. The calculator is now asking for a Guess. By this question the calculator is asking us to move the highlight toward the point of intersection.
Figure 19	We can accomplish that movement by using, repeatedly, the key in this case. If we do this we can get to the image shown in Figure 19. We are telling the calculator that the highlighted point is in the direction toward the point of intersection. Now we just need to press the key to have the calcualtor actually find that point of intersection.
Figure 20	In Figure 20 the calculator has done all the work and it teslls us that the point f Intersection is when x=8 and y^–3.

As we have done in other pages, we will now look at a second problem.

Solve:

3x + 5y	=	47	(5)
4x + 11y	=	93	(6)

To get the equations into the calculator we need to solve them for y. Thus, our two equations become:

y	=	(1x + 47) / 5	(7)
y	=	(^–4x + 93) / 11	(8)

Figure 21	Figure 21 shows that we have changed the Y= screen to hold our new equations. [Remember that we get to that screen via the key sequence.]
Figure 22	Again,press to get to the ZOOM window shown in Figure 22. Again we will select the Standard option by pressing the key. This will take us to Figure 23.
Figure 23	The graph shown in Figure 23 shows the two functions. Unfortunately, the slopes of the two functions are not that far apart so the overlap on the TI-3/84 qindow makes it really hard to see just where the two funtions cross.
Figure 24	For Figure 24 we use the key to return to the WINDOW screen. In Figure 23 we can see that we want to look at the first quadrant, I, so we will change the various values in Figure 24 to reflect that. Those changes are shown in Figure 25.
Figure 25	Once the changes are made, as shown here, we press the key to draw another graph.
Figure 26	Figure 26 shows the new graph. On this graph, Figure 26, it is a little easier to at least guess where the two lines cross. Still, it is not worth the struggle. We can let the calculator find thepoint of intersection. Use the key sequene to open the CALC screen.
Figure 27	Of the items on the CALC screen we want the fifth item, intersect. THerefore, press the key to move to Figure 28.
Figure 28	The calculator has placed the highlighter on the first function. This is fine with us so we press the key to select that as one of the lines to use.
Figure 29	Now, in Figure 29, the calculator proposes using the second line. We agree so we again press .
Figure 30	Having selected the two lines to use, the calculator wants us to move toward the point of intersection and then use that new location as a first guess. Experience in using the calculator for such cases as the intersection of two lines has shown that we just have to press here. That will start the calculator on its effort to find the point of intersection.
Figure 31	In Figure 31 the calculator has given us the point of intersection, namely, where x=4 and y=7, or the point (4,7).