Meet You at the Intersection: Solving a System of Linear Equations

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1 Meet You at the Intersection: Solving a System of Linear Equations Activity 30 Many times, the solution to a real-life problem involves solving more than one mathematical equation at the same time. The simplest situation of this type involves a pair of equations with two unknown quantities called a linear system. Graphically, the solution to this kind of system represents the point where the graphs of these two lines intersect. For example, imagine that a person is running to catch up to his friend who is walking ahead of him. These motions can be modeled graphically by plotting distance versus time. The motion graphs are linear if each person is moving at a constant speed. If the plots are made on the same set of axes, the point where the two lines cross represents the physical location where the two friends pass each other. It is possible to model situations of this type in the classroom using two Motion Detectors. In this activity you will collect and analyze motion data in order to determine the solution to a linear system of equations. OBJECTIVES Record motion data for two walkers. Graph both motions on a common axis and find their intersection. Find linear equations to model the motions. Solve the system of two linear equations to determine the intersection. Compare the algebraic solution to the graphical solution. Real-World Math Made Easy 2005 Texas Instruments Incorporated 30-1

3 Meet You at the Intersection 4. You will need two people to walk, a third (the timer) to operate the stopwatch, and a fourth (the marker) to mark the place where the two walkers pass each other. Practice walking in the following manner: Walker one stands about two meters from the first detector, and prepares to walk directly toward that detector. Walker two stands just over half a meter from the second detector, and prepares to walk directly away from that detector. Once the timer has started data collection, timing must also be started. At the moment the two walkers pass one another, the marker must note the position on the floor, while the timer stops the stopwatch. The walkers then continue to walk at a slow but uniform pace until data collection ends. Walker one must never get closer than 0.5 m to the Motion Detector. Practice the procedure several times. 5. Select (on both calculators simultaneously if you are using two) to begin data collection. Walk as you practiced. Data collection will run for ten seconds. 6. When data collection is complete, a graph of distance versus time will be displayed on the calculator screen(s). Examine the distance versus time graphs. Walker one s graph should be a nearly linear and uniformly decreasing function. Walker two s graph should be a nearly linear and uniformly increasing function. If the final few seconds of data are not useful, do not be concerned. Check with the teacher if you are not sure whether you need to repeat the data collection. To repeat data collection, select and repeat Step Once you are satisfied with the graph, select to return to the Main screen. Exit EasyData by selecting from the Main screen and then selecting. 8. After a good run, have the marker use the meter stick to measure the distance from the point where the two walkers passed one another to the midpoint of the line between the two Motion Detectors. Record this directly measured distance in the Data Table on the Data Collection and Analysis sheet. (Round this and all other data to three significant figures.) Record the directly measured time interval determined by the timer in the Data Table. ANALYSIS 1. (Skip this step if you are using LabPro with one calculator.) To complete the analysis, you need to move the distance data from one calculator to another. In order to avoid overwriting data, you will first copy the data to a new location in the calculator, and then send it to the receiving calculator. Use walker one s calculator as the sending calculator, and walker two s as the receiving calculator. Be sure you know which is which! a. Disconnect the calculators from the data-collection interfaces, and directly connect the two calculators with a black link cable. Firmly press in the cable ends. b. On the sending calculator, copy the distance data to a new list name. To do this, press [L6] [L2]. Real-World Math Made Easy 2005 Texas Instruments Incorporated 30-3

4 Activity 30 c. You will now have the expression L6 L2 on the home screen. Press to copy the distance data in L6 to L2. d. Press [LINK]. e. Press the number adjacent to List to select a data list. f. Press until L2 is highlighted. Press to select it. g. Press to highlight TRANSMIT. h. On the receiving calculator, press [LINK]. i. Press to highlight RECEIVE, and press. The receiving calculator will show Waiting j. On the sending calculator, press to actually transmit the list. k. The receiving calculator may display a message screen headed DuplicateName. If it does, press the number adjacent to Overwrite. The receiving calculator will show Done. 2. Perform the following to display the two distance graphs together. Use the receiving calculator if you completed Step 1. a. Press [STAT PLOT] and press to select Plot 1. b. Change the Plot1 settings to match the screen shown here. Press to select any of the settings you change. c. Press [STAT PLOT] and press to select Plot 2. d. Change the Plot2 settings to match the screen shown here. Press to select any of the settings you change. (If you are using LabPro with one calculator, use L9 instead of L2. To enter L9, press [LIST], scroll to L9, and press to paste it to the stat plot screen.) e. Press and then select ZoomStat (use cursor keys to scroll to ZoomStat) to draw a graph with the x and y ranges set to fill the screen with data. f. Press to determine the coordinates of a point on the graph using the cursor keys. Answer Question 1 on the Data Collection and Analysis sheet. 3. Trace across either line to the place where the two lines cross. Record the coordinates as the Graphical Trace result in the Data Table on the Data Collection and Analysis sheet. 4. To create two linear models for the data, you need to record two points from each line. Ignore any short horizontal sections at the beginning or end of the lines. Press to move from one line to the other. Trace across the graphs and record the coordinates of two well-separated points from each line. Record these values in the Data Table. 5. Find the slope of each line using the standard slope formula, and enter the slopes in the Data Table. y2 y1 m = x x Texas Instruments Incorporated Real-World Math Made Easy

6 Activity 30 DATA COLLECTION AND ANALYSIS Name Date DATA TABLE Direct Measure Graphical Trace Calculator Intercept Search Solve Equation System Intersection Time Intersection Position Walker One Walker Two x 1, y 1 x 2, y 2 Slope Intercept QUESTIONS 1. How can you identify which trace is walker one and which is walker two? Identify the trace of each walker for future reference. 2. How do the model lines fit the walker data? Real-World Math Made Easy 2005 Texas Instruments Incorporated 30-6

7 Meet You at the Intersection 3. Compare the values you obtained for the intersection time by all four methods: (1) direct measurement by stopwatch and meter stick; (2) graphical intersection by tracing, (3) the calculator s intersection search, and (4) by solving the system of equations. Are the values consistent? 4. Is it possible for the walkers to move in front of the Motion Detectors so that the resulting plots would not intersect? If so, give an example; if not, explain why. 5. Could the walkers move so that their plots intersect more than once? If so, how? Real-World Math Made Easy 2005 Texas Instruments Incorporated 30-7

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