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					                        Linear Equations
The Mathematics Readiness Project was funded jointly by the Eisenhower State
Grant Program in Mathematics and Science and the California Academic
Partnership Program.
                                                             Overview for the Teacher


                      Mathematics Readiness Projec t
                            Linear Equations
                                     by Katrine Czajkowski

Overview for the Teacher

Consider the following question:

Which of the following is a portion of the graph of y = –2x + 4?

(a)                                  (b)                     (c)




(d)                                  (e)




The incorrect answer most commonly chosen is (e). Among the reasons
students might miss this question are the following:

1.    Students might not realize that y = –2x + 4 describes the relationship
      between two variables.
2.    Students might not recognize that the equation is linear.
3.    Students might not recognize a line graphed on the Cartesian plane as the
      graphical representation of a linear equation.
4.    Students might reverse the slope and y-intercept terms.
5.    Students might not understand how to identify the y-intercept of a line.
6.    Students might not understand that slope is simply a way to express the
      ratio of the vertical change vs. the horizontal change.
7.    Students might not be able to connect the direction of the line drawn to
      the sign of its slope.
8.    Students might not distinguish between a slope of 2 and a slope of –2.

Mathematics Readiness Project 1997
The misunderstandings described above can become particularly acute when
students face a more complex question such as

    x+y=2 
If         , then y =
    x–y=6 

(a) –8                (b) –4         (c) –2   (d) 2   (e) 8

The incorrect answer most commonly chosen is (b).

This unit is intended to provide teachers with ways to supplement their
curriculum in order to better prepare students to succeed in their study of
algebra.




Mathematics Readiness Project 1997                                     Page 2
                                     Table of Contents


Section 1: Characteristics of Linear Equations

Section 2: Tables of Values for Graphing

Section 3: Using Intercepts for Graphing

Section 4: Using Slope-Intercept Form with Graphs

Section 5: Families of Linear Equations

Section 6: Graphing Linear vs. Non-Linear Equations

Section 7: Creating, Graphing and Using Linear Equations

Section 8: Simple System of Equations

Section 9: What Went Wrong?

Section 10: Exploring with a Graphing Calculator




Mathematics Readiness Project 1997                         Page 3
Section 1: Characteristics of Linear Equations                      Teacher Page


For the purpose of this lesson, we will not discuss constant linear equations
(such as x = 7 or y = –10), as they offer exceptions to the rules governing most
linear equations.

Explain to the students that linear equations have several basic characteristics:

1)    The expression contains an equals (=) sign.
2)    The expression contains two variables (usually denoted by "x" and "y").
3)    The expression can be manipulated using addition or subtraction so that
      it appears like either y = mx + b or cx + dy = r, where b, c, d, m and r are
      constant.

Work through the following examples with the students:

Consider the equation y = 2x + 3. Is this equation linear?

Analysis
1) Does the expression contain an equals (=) sign? Yes.
2) Does the expression contain two variables? Yes.
3) Can the expression be manipulated using addition or subtraction so that it
   appears like either y = mx + b or cx + dy = r, where b, c, d, m and r are
   constant? Yes, it already looks like y = mx + b.

Is the equation linear? Yes.

Make a table of values where x always changes by the same amount. One
possible response is

                                 x   –2   –1   0   1   2     3
                                 y   –1   1    3   5   7     9

Notice that when the values for x change by the same amount, the values for
y also always change by the same amount.




Mathematics Readiness Project 1997                                            Page 4
Section 1: Characteristics of Linear Equations                   Teacher Page



Consider the equation xy = 8. Is this equation linear?

Analysis
1) Does the expression contain an equals (=) sign? Yes.
2) Does the expression contain two variables? Yes.
3) Can the expression be manipulated using addition or subtraction so that it
   appears like either y = mx + b or cx + dy = r, where b, c, d, m and r are
   constant? No.

Is the equation linear? No.

Make a table of values where x always changes by the same amount. One
possible response is

                                 x   1   2   3   4
                                 y   6   3   2   1.5

Notice that the values for y do not change by the same amount.


The problems in the student pages of Characteristics of Linear Equations
reinforce these ideas. Students may work individually, in pairs or in groups
of no more than four. Be sure to review their work with them.

Answers to student page questions:

I. The equation is linear.
II. The equation is linear.
III. The equations y = 3x + 3, 2y + 5x = 10,
     4 = x – 2y and 2x – 3y = 12 are linear.




Mathematics Readiness Project 1997                                       Page 5
Section 1: Characteristics of Linear Equations                          Student Page


I.    Given the equation y = 4 – x, complete the table below.

•     Under "yes/no," check whether or not the equation meets each
      characteristic in the first column.

•     In the final column, provide evidence to support your choice of either
      "yes" or "no."

Be sure to complete the table of values box. Choose values for x that always
change by the same amount.

In the box at the bottom of the table, answer the question and provide a one-
sentence reason for your answer.

Characteristics of linear equations              Evidence to support your decision
1.    Does     the    expression
      contain an equals (=) sign?

2.    Does     the    expression
      contain two variables?


3.    Can the expression be
      manipulated            using
      addition or subtraction so
      that it appears like either y
      = mx + b or cx + dy = r,
      where b, c, d, m and r are
      constant?




                                      x
                                      y


Is this equation linear?




Once you have completed this table and reviewed it with your teacher,
proceed to the next page and complete the chart based on the expression
given.




Mathematics Readiness Project 1997                                                   Page 6
Section 1: Characteristics of Linear Equations                          Student Page


II. Given the equation x – y = 8, complete the table.

•     Under "yes/no," check whether or not the equation meets each
      characteristic in the first column.

•     In the final column, provide evidence to support your choice of either
      "yes" or "no."

Be sure to complete the table of values box. Choose values for x that always
change by the same amount.

In the box at the bottom of the table, answer the question and provide a one-
sentence reason for your answer.

Characteristics of linear equations              Evidence to support your decision
1.    Does     the    expression
      contain an equals (=) sign?

2.    Does     the    expression
      contain two variables?


3.    Can the expression be
      manipulated            using
      addition or subtraction so
      that it appears like either y
      = mx + b or cx + dy = r,
      where b, c, d, m and r are
      constant?




                                      x
                                      y


Is this equation linear?




Once you have completed this table and reviewed it with your teacher,
proceed to the next page and complete the chart based on the expression
given.




Mathematics Readiness Project 1997                                                   Page 7
Section 1: Characteristics of Linear Equations                               Student Page


III. Now complete the table below. Check either "linear" or "not linear" for
each equation. In the final column, provide evidence to support your
decision.

 Expression             Linear?      Not linear?   Evidence to support your decision

  y = 3x + 3




 2y + 5x = 10




   3xy = 12




   4 = x – 2y




   x2 + 2 = y




 2x – 3y = 12




Mathematics Readiness Project 1997                                                     Page 8
Section 2: Tables of Values for Graphing                       Teacher Page


Provide the students with a brief review of plotting points in the Cartesian
plane. They then should be able to work through the student pages of Tables
of Values for Graphing. Graph paper should be available for the students.

Answers to student page questions:
I.




II.




III.




Mathematics Readiness Project 1997                                      Page 9
Section 2: Tables of Values for Graphing                               Student Page


I.    Consider the equation y = –2x + 2.

      Step 1: Make an (x,y) table using at least three of your favorite values for x.
      Pick at least one positive value for x, at least one negative value for x, and
      use zero as a value for x.


                                 x

                                 y

      Step 2: Plot the points on the Cartesian plane.




      Step 3: In the plot above, draw a line running through all the points.




Mathematics Readiness Project 1997                                              Page 10
Section 2: Tables of Values for Graphing                        Student Page


For each equation below, construct the (x,y) table of values and graph the
equation on the Cartesian plane. Work with a partner and follow the steps for
using a table of values to graph a linear equation. Show your work in the
space provided.

II. y = 3x – 2


        x

        y




III. 2x + y = 8


        x

        y




Mathematics Readiness Project 1997                                      Page 11
Section 3: Using Intercepts for Graphing                     Teacher Page


Work through the graphing of
3x + 4y = 12 with the students.
Do this by first identifying the
x-intercept—(4,0)—and the y-
intercept—(0,3)—and         then
drawing the line that runs
through these two points.




Then     work     through   the
somewhat       more     complex
                         1
problem of graphing y = 2 x – 4
by using the x-intercepts and
the y-intercepts.




Have the students work through the Using x-Intercepts and y-Intercepts for
Graphing student pages.




Mathematics Readiness Project 1997                                   Page 12
Section 3: Using Intercepts for Graphing     Teacher Page




Answers to student page questions:

I.                                     II.




III.                                   IV.




Mathematics Readiness Project 1997                  Page 13
Section 3: Using Intercepts for Graphing                                 Student Page


The x-intercept is the point where the line crosses the x-axis.

The y-intercept is the point where the line crosses the y-axis.

The two intercepts can be used to quickly graph a linear equation.

I.    Follow the procedure below to graph the equation y = 2x + 4.

      Step 1: Find the y-intercept: In the equation substitute the value 0 for x
      and solve for y.


      Step 2: Fill in the missing number: (0,    ) is the y-intercept of y = 2x + 4.

      Step 3: Find the x-intercept: In y = 2x + 4 substitute the value 0 for y and
      solve for x.



      Step 4: Fill in the missing number: (     ,0) is the x-intercept of y = 2x + 4.

      Step 5: Plot the x-intercept
      and the y-intercept on the
      Cartesian plane.




      Step 6: On the graph above, draw a line running through the two
      intercepts.




Mathematics Readiness Project 1997                                                 Page 14
Section 3: Using Intercepts for Graphing                                  Student Page


For each of the equations below,

a)    Identify the x-intercept of the graph by substituting zero for y.
b)    Plot the x-intercept on the Cartesian plane.
c)    Identify the y-intercept of the graph by substituting zero for x.
d)    Plot the y-intercept on the Cartesian plane.
e)    Draw a line running through the two intercepts.

II. y = 3x – 1

                                                         x-intercept: (       ,     )



                                                         y-intercept: (       ,     )

                                                   Coordinates of third point:

                                                               (   ,      )

                                                   Does this point lie on the line
                                                   you drew after finding the x-
                                                   and y-intercepts?


                                                   Why is the          last       question
                                                   important?




Mathematics Readiness Project 1997                                                      Page 15
Section 3: Using Intercepts for Graphing                         Student Page


III. x – 2y = 8

                                                x-intercept: (       ,     )



                                                y-intercept: (       ,     )

                                           Coordinates of third point:

                                                      (   ,      )

                                           Does this point lie on the line
                                           you drew after finding the x-
                                           and y-intercepts?


                                           Why is the         last       question
                                           important?




Mathematics Readiness Project 1997                                             Page 16
Section 3: Using Intercepts for Graphing                         Student Page


IV. x = 4 + y

                                                x-intercept: (       ,     )



                                                y-intercept: (       ,     )

                                           Coordinates of third point:

                                                      (   ,      )

                                           Does this point lie on the line
                                           you drew after finding the x-
                                           and y-intercepts?


                                           Why is the         last       question
                                           important?




Mathematics Readiness Project 1997                                             Page 17
Section 4: Using Slope-Intercept Form with Graphs               Teacher Page


In this section you should present to the students the ideas of slope and y-
intercept. Below we outline for you a possible presentation of these concepts.

Demonstrate for the students
that in a table of values for the
equation y = 2x + 5, the y values
increase by 2 as the x values
increase by 1. Show also that for
any two points on this line, the
ratio of the vertical change
divided by the horizontal
change is 2. Define this number
to be the slope of the line.




Use the equation y = –3x + 1 to
graphically illustrate the idea of
negative slope (–3 in this case).

Point out to the students that
the line represented by the
equation y = –3x + 1 crosses the
y-axis at the point (0,1). This is
called the y-intercept.




Mathematics Readiness Project 1997                                       Page 18
Section 4: Using Slope-Intercept Form with Graphs               Teacher Page


Use the graph of the equation
y = 2x + 5 to reinforce the idea
of y-intercept. In this case, the
y-intercept is at (0,5)




Summarize for the students the idea that the line represented by an equation
of the form y = mx + b has slope "m" and y-intercept (0,b). We call y = mx + b
the slope-intercept form of the equation of the line.

Have the students work through the Using Slope-Intercept Form with Graphs
student pages. Graph paper should be available for the students.




Mathematics Readiness Project 1997                                       Page 19
Section 4: Using Slope-Intercept Form with Graphs     Teacher Page



Answers to student page questions:

I.    y = –x + 5                     II. y = 2x – 4




         1
III. y = 2 x + 2




IV. y = –2x – 6

       2
V. y = 3 x + 2

VI. y = 3x – 5




Mathematics Readiness Project 1997                           Page 20
Section 4: Using Slope-Intercept Form with Graphs                   Student Page


In problems I through III, rearrange the terms in the given equation so that
each equation appears in slope-intercept form, y = mx + b. Then graph the
equation by using the y-intercept and the slope.

Remember that

•     The "y" must be alone and positive.

•     The "b" is a constant that is not attached to either variable. The "b"
      represents the y-intercept of the line.

•     The "m" is the coefficient of x and represents the slope of the line. The
      slope is the ratio of the vertical change to the horizontal in the line from
      one point to another.

1.    x+y=5                          Slope-intercept form:




Mathematics Readiness Project 1997                                           Page 21
Section 4: Using Slope-Intercept Form with Graphs            Student Page


II. 4x – 2y = 8                      Slope-intercept form:




III. 3x = 6y – 12                    Slope-intercept form:




Mathematics Readiness Project 1997                                  Page 22
Section 4: Using Slope-Intercept Form with Graphs                Student Page


In problems IV through VI, find a linear equation for the line shown. Begin
by finding the y-intercept ("b"). Then use the two given points to identify the
slope ("m"). Finally, write the equation in slope-intercept form.

IV.                                    b=



                                       m=



                                       Equation:




V.                                     b=



                                       m=



                                       Equation:




VI.                                    b=



                                       m=



                                       Equation:




Mathematics Readiness Project 1997                                        Page 23
Section 5: Families of Linear Equations                          Teacher Page


Common errors in graphing linear equations often result from

1.    Confusing the slope (m) with the y-intercept (b).
2.    Inadvertently reversing the sign of the slope.
3.    Plotting the y-intercept on the wrong axis.

If students are alerted to these common errors, they will be better prepared to
avoid them. Studying families of related equations may raise students'
awareness of these potential errors.

Show the students how the
graphs of

                   y = 2x + 4

and

                  y = –2x + 4

 differ. This will help them
recognize the significance of
the "m" term.




Show the students how the
graphs of

                   y = 2x + 4

and

                   y = 2x – 4

differ. This will draw their
attention to the significance of
the "b" term.




Mathematics Readiness Project 1997                                        Page 24
Section 5: Families of Linear Equations                        Teacher Page


Show the students how the
graphs of

                   y = 2x + 4

and

                   y = 4x + 2

differ. This will alert them to
the problems associated with
confusing the slope of a line
with its y-intercept.




The comparison charts in the Families of Linear Equations student pages will
help reinforce these important concepts. Students should be encouraged to
create their own comparison charts based on original equations.




Mathematics Readiness Project 1997                                     Page 25
Section 5: Families of Linear Equations   Teacher Page


Answers to student page questions:

I.




II.




III.




Mathematics Readiness Project 1997               Page 26
Section 5: Families of Linear Equations                                        Student Page


For each of the pairs of equations below, complete the chart. Indicate the slope
and the y-intercept of each equation and graph each line.

Answer the question beneath the graphs using at least one complete sentence.

I.

Equation:                            y = 3x + 5                       y = –3x + 5


Slope:


y-intercept:




Graph:




What is the reason for the difference in the graphs of the two equations?




Mathematics Readiness Project 1997                                                    Page 27
Section 5: Families of Linear Equations                                         Student Page


II.

Equation:                            y = 2x + 3                        y = 2x – 3


Slope:


y-intercept:




Graph:




What is the reason for the difference in the graphs of the two equations?




Mathematics Readiness Project 1997                                                     Page 28
Section 5: Families of Linear Equations                                         Student Page


III.

Equation:                            y = –3x + 2                       y = 2x – 3


Slope:


y-intercept:




Graph:




What is the reason for the difference in the graphs of the two equations?




Mathematics Readiness Project 1997                                                     Page 29
Section 6: Graphing Linear vs. Non-Linear Equations                  Teacher Page


Use the equation x2 – y = 4 to show the students that not all graphs are lines.
Proceed as follows:

First, generate a table of values for x2 – y = 4:

                                 x   –3   –2   –1    0    1   2     3
                                 y    5    0   –3   –4   –3   0     5

Second, point out that although the x values are increasing by the same
amount, namely 1, the y-values are changing by different amounts. For
example, as x increases from 0 to 1 y increases by 1, while when x increases
from 1 to 2 y increases by 3.

Third, plot the points so that the students will see that the graph is not a line:




Ask students if they know of other characteristics that would indicate that the
graph would not turn out to be a line.

Having the students work through the student pages of Graphing Linear vs.
Non-Linear Equations will reinforce the fact that not all graphs are lines.




Mathematics Readiness Project 1997                                            Page 30
Section 6: Graphing Linear vs. Non-Linear Equations   Teacher Page


Answers to student page questions:

I.    The equation –3x + y = 6 is linear.




II. The equation 2xy = 8 is not linear.




III. The equation 4y = 2x is linear.




Mathematics Readiness Project 1997                           Page 31
Section 6: Graphing Linear vs. Non-Linear Equations             Student Page


For each equation,
    a) Solve for y in the first box.
    b) Create a table of values based on the equation.
    c) Graph the equation.
    d) Indicate whether or not the equation is linear by writing "yes" or "no"
       in the first box.


              Equation               Table of Values         Graph


              –3x + y = 6               x     y




Linear?




                2xy = 8                 x     y




Linear?




                4y = 2x                 x     y




Linear?




Mathematics Readiness Project 1997                                       Page 32
Section 7: Creating, Graphing and Using Linear Equations            Teacher Page


It is important for students to recognize that equations are just ways of
describing the special relationship that exists between two variables. It is
therefore helpful to provide students with real-world applications of these
ideas.

Work through the following application with the students:

Tickets for the school play go on sale Monday. Prior to this a total of 27 tickets
have been purchased by members of the cast for their families. As the day
progresses a total of 6 tickets are sold each hour. Thus, if "h" represents the
number of hours that have gone by and if "t" represents the number of tickets
that have been sold, we can create a table of values:

                                 h    0    1    2    3    4
                                 t   27   33   39   45   51

We also can write an equation that represents this situation: t = 6h + 27

Finally, we graph this
equation. Draw the
graph only for h
between 0 and 6.
Point out that the
graph is a straight
line.




Show how to use the graph to estimate how many tickets will be sold after 10
hours. [The answer 87]

The problems in the student pages of Creating, Graphing and Using Linear
Equations provide the students with other applications of linear equations.




Mathematics Readiness Project 1997                                           Page 33
Section 7: Creating, Graphing and Using Linear Equations   Teacher Page


Answers to student page questions:

I.    The equation is B = 15t.




      Monique will have 15 tests graded by the
      time She has $225 in her bank account.

II. The equation is I = 5s + 200.




      Mr. Gonzalez's income will be $250 if he
      sells 10 shares of stock.




Mathematics Readiness Project 1997                                Page 34
Section 7: Creating, Graphing and Using Linear Equations            Student Page


For each situation below,
    a) Write an equation that describes the relationship between the two
       variables mentioned. State whether or not the equation is linear.
    b) Draw the graph of each equation in the space provided.
    c) Write two statements that are true based on your graph.
    d Use the graph to estimate the amount requested in the final statement.

I.    Monique earns $15 for every biochemistry test she grades as a teacher's
      assistant. She starts the semester with nothing in the bank and saves all of
      the money he earns grading tests. Use "B" to represent the amount of
      money in her bank account and "t" to represent the number of tests she
      grades.

      Equation:



      Statements based on the graph:




      Using your graph, estimate how
      many tests Monique will have
      graded by the time she has $225
      in her bank account.




Mathematics Readiness Project 1997                                           Page 35
Section 7: Creating, Graphing and Using Linear Equations       Student Page


II. Each week, Mr. Gonzalez earns $200 per week plus $5 for every share of
    stock he sells. His income is represented by "I" and the amount of stock
    he sells is represented by "s."

      Equation:



      Statements based on the graph:




      Using your graph, estimate Mr.
      Gonzalez's income if he sells 10
      shares of stock.




Mathematics Readiness Project 1997                                     Page 36
Section 8: Simple System of Equations                                           Teacher Page


Work through the following application with the students:

Thomas has been offered a job where he can earn $5 for every hour he spends
painting houses. If he starts with nothing in his bank account and doesn't
spend any money all week, how much will be in his bank account?

The answer obviously depends upon how many hours Thomas works. If "h"
represents the number of hours he works and "a" represents the amount of
money in his bank account, we know that a = 5h

Suppose now that Thomas has a alternate job offer in which he would be
given an initial payment of $12 and then would earn $3 an hour? Which job
would give Thomas the most money after 10 hours? After 10 hours, the first job
would give him 5× 10 = $50 and the second job would give him only 3 × 10 + 12 = $42.

After how many hours of work will both jobs pay him the same amount?

      Complete the following table for the first equation, a = 5h:

      h           0        1         2    3    4    5    6    7    8    9

      a           0        5         10   15   20   25   30   35   40   45

      Complete the following table for the second equation, a = 3h + 12:

      h           0        1         2    3    4    5    6    7    8    9

      a           12      15         18   21   24   27   30   33   36   39

Point out that the pair (6,30) appears in both tables and that 6 hours of work
would pay him the same amount from both jobs.

Draw the graphs of
both equations on the
same axes and point
out that (6,30) is the
point of intersection.

Discuss which job
pays more after 10
hours.




Students should now complete the student pages of Simple Systems of
Equations.

Mathematics Readiness Project 1997                                                     Page 37
Section 8: Simple System of Equations   Teacher Page


Answers to student page questions:

I.    (5,3)




II. (1,5)
III. (2,4)
IV. (–1,6)




Mathematics Readiness Project 1997             Page 38
Section 8: Simple System of Equations                                                   Student Page


I.    Consider the system of equations

                                                      x–y=2
                                                               
                                                                
                                                               
                                                     
                                                      x+y=8    
                                                                

      Complete the following table for                         Complete the following table for
      the first equation, x – y = 2:                           the second equation, x + y = 8:

      x          1    2     3        4   5   6   7             x    1   2   3   4   5   6   7

      y                                                        y



      What is the common pair (x,y) on both tables?


      Verify that this pair (x,y) satisfies the first equation, x – y = 2.




      Verify that this pair (x,y) satisfies the second equation, x + y = 8.




      To the right, draw the
      graphs of both equations,
      x – y = 2 and x + y = 8.




      At what point (x,y) do the
      two lines intersect?



      Explain in a full sentence
      how this point (x,y) is
      related to the (x,y) found
      above.




Mathematics Readiness Project 1997                                                              Page 39
Section 8: Simple System of Equations                          Student Page


In problems II, III and IV, make a table of values for each of the two given
equations to answer the question.

        x+y=6 
II. If              , then y =
        y = 3x + 2 




         y=x+2 
III. If              , then x =
         y = 3x – 2 




        2x + y = 4 
IV. If              , then y =
        y = 3x + 9 




Mathematics Readiness Project 1997                                     Page 40
Section 9: What Went Wrong?                                  Teacher Page


Students can avoid many future mistakes if they have the opportunity to
critique the work of others, especially if that work contains frequently
encountered errors. In this section the students are asked to describe the
mistake and write the correct answer.

Answers to student page questions:

(c) is the correct answer.




Mathematics Readiness Project 1997                                   Page 41
Section 9: What Went Wrong?                                        Teacher Page


I.    Amy tried to answer the following question:

Which of the following is a portion of the graph of y = –2x + 4?

(a)                                  (b)             (c)




(d)                                  (e)




She wrote (e) for her answer. Explain what Amy did wrong. Use complete
sentences.




What is the correct answer?



II. Max wrote (a) for his answer to the same question. Explain what Max did
    wrong. Use complete sentences.




Mathematics Readiness Project 1997                                        Page 42
Section 10: Exploring with a Graphing Calculator               Teacher Page


If you have graphing calculators available, then students can use them to to
find the intersection of two graphs. Lead the class through the following
graphing calculator activities:

1.    Graph y = 2x – 3 on
      the           graphing
      calculator. By tracing
      verify that the y-
      intercept is at (0,–3)
      and that the x-
      intercept is at (1.5,0).




2.    If    the     graphing              x         y
      calculator     has    a            –1         5
      "table" option, use                 0        –3
      the table to view a                 1        –1
      listing of (x,y) values             2         1
      for y = 2x – 3. Use                 3         3
      these     values     to             4         5
      verify that the slope
      of the line is 2.



3.    Graph y = –3x + 2
      now on the graphing
      calculator,    leaving
      the     first    graph,
      y = 2x – 3, "on."




Mathematics Readiness Project 1997                                     Page 43
Section 10: Exploring with a Graphing Calculator                       Teacher Page


4.    By     tracing      and
      zooming, verify that
      the      point        of
      intersection is (1,–1).




5.    If the graphing calculator has a "table" option, use the table to verify that
      the coordinates (1,–1) appear on both tables.

                          x          y = 2x – 3    x     y = –3x + 2
                         –1               5       –1           5
                          0              –3        0           2
                          1              –1        1          –1
                          2               1        2          –4
                          3               3        3          –7
                          4               5        4         –10


Students should now complete the student pages of Exploring with a
Graphing Calculator.




Mathematics Readiness Project 1997                                            Page 44
Section 10: Exploring with a Graphing Calculator   Teacher Page


Answers to student page questions:

III. (1,2)




IV. (–4,–4)




Mathematics Readiness Project 1997                        Page 45
Section 10: Exploring with a Graphing Calculator                  Student Page


Use your graphing calculator to help do the following problems.

I.    Graph y = –x + 3.

      What is the x-intercept? Label
      it on your graph.


      What is the y-intercept? Label
      it on your graph.


      Give the coordinates of three
      other points that lie on this
      line:

      (      ,      ), (        ,    ), (   ,   )


      What is the y-coordinate of the
      point with x-coordinate –4?



II. Turn off the graph for #I.
    Graph –3x + y = –1.

      What is the x-intercept? Label
      it on your graph.


      What is the y-intercept? Label
      it on your graph.


      Give the coordinates of three
      other points that lie on this
      line:

      (      ,      ), (        ,    ), (   ,   )




Mathematics Readiness Project 1997                                       Page 46
Section 10: Exploring with a Graphing Calculator                  Student Page



III. Turn on the graph for #I,
     namely y = –x + 3.

      On this set of axes, sketch the
      graph of y = –x + 3 and the
      graph of –3x + y = –1

      What are the coordinates of
      the point where the two lines
      intersect?
                ( ,   )

      Use "table" mode to view (x,y)
      tables for each graph. What
      pair of coordinates appears in
      the table for both graphs?

                       (     ,       )

                                                              1
                                                       
                                                        y =2 x – 2  
IV. Use a graphing calculator to graph the equations              5   and
                                                        6–y=–2x 
                                                                     
    then use the graphs and the calculator functions to identify the solution
    to the system.




V. Imagine that your friend has called you on the phone to ask for help
   using a graphing calculator to find the solution to a system of two
   equations with two variables. On separate paper, write out your
   explanation. Remember, your friend cannot see your calculator over the
   phone, so your directions must be very specific and clear.




Mathematics Readiness Project 1997                                       Page 47
Section 10: What Went Wrong?         Student Page




Mathematics Readiness Project 1997          Page 48

				
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