MPM2D Activity 3.1: Analyzing Systems of Equations

Specific Expectation:

Determine the point of intersection of two linear relations graphically, with and without the use of graphing calculators, and interpret the meaning of the intersection point.

PART A: SOLVE BY HAND

Ali Gator’s father is planning to buy a cellular phone, and Ali is helping him decide which one to buy by investigating different service providers and the plans they have. This is what she has found:

  1. Which plan do you think Ali should recommend? Explain your reasons.

    2.  

     

     

  2. Complete the table of values below, that compares the number of minutes of calling time used in a month to the total charge.

Number of Minutes used

Total Charges for Cantel ($)

Total Charges for Rogers ($)

0

$20.00

$25.00

10

    

20

    

30

$24.50

  

40

    

50

    

60

    

70

    

80

    

90

    

100

    

3. Create the equation that describes each of these relations.

Cantel: ___________________

Rogers: ___________________

4. Construct, by hand, (on graph paper) a graph of the relation that compares the number of minutes used with the total charges for the Cantel plan. Then, on the same set of axes, graph the same relation for the Rogers plan. Should the points be connected? Explain.

 

 

 

5. Looking at the lines you have just graphed, which plan do you think Ali should recommend, and why? Under what circumstances is one plan better than the other? Does Ali need any more information before she makes her recommendation?

 

 

 

 

 

6. Circle the point on the graph where both plans have the same total charges. Estimate the coordinates of this point.

PART B: USING THE GRAPHING CALCULATOR

Before you begin, clear out any existing functions and turn off all StatPlots.

  1. Identify the independent variable: ______________________

    2. and the dependent variable: _______________________.

    Replace the independent variable with X and the dependent variable with Y so the formula should now be in the form "Y="

  2. GRAPH the two relations on your graphing calculator. Remember that in order to do this, you need to enter the formula and set the WINDOW. Use your paper graph to choose the best values for the minimum and maximum values.
  3. METHOD 1: Using the arrow keys, TRACE the graph to the point of intersection of the two lines. Record the coordinates below, to two decimal places.

    4. x-coordinate: ______________

    y- coordinate: ______________

    How close are these coordinates to your estimates in Part 1? Why?

     

     

     

  4. ZOOM in on the point of intersection to get a better reading of the coordinate. Record below your best reading, to two decimal places.

    5. X-coordinate: ______________

    Y- coordinate: ______________

  5. METHOD 2: The graphing calculator can also construct table of values. Using the TBLSET key, start the table at 0 and have the calculator go up automatically by increments of 10. Press TABLE to view it. Notice that the numbers in the second column are all lower than the corresponding numbers in the third column. What does this say about the two cell phone plans?

    6.  

     

  6. Scroll down until you find the place in the table where the numbers in the second column begin to be larger than those in the third column. Where dos this happen?

    7.  

  7. We can narrow in on the exact place where the switch happens by changing the TBLSET. This time, start your table at 70 and have the numbers go up by ones instead of tens. Continue to narrow in until you can specify the exact place where the switch happens to two decimal places. Record the numbers in all three columns.

X: __________ Y1: __________ Y2: __________

9. METHOD 3: The calculator has a function to determine the point of intersection between two lines. Located above the trace button is CALC. In this method, the command INTERSECT can be used. Use this command and determine the point of intersection.

X: __________ Y: __________

10. Ali has found two other cellular phone plans for her father to consider:

Find the equation that relates the number of minutes to the total monthly charge, for each plan. Graph the relations and make a recommendation about the two phone plans. Which plan is better?

Fido: __________________

AT&T: __________________
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