Activity 3 - 1

1
Activity 3 - 1

• Business Checking Account

2
Objectives

• Solve a system of two linear equations
  numerically
• Solve a system of two linear equations
  graphically
• Solve a system of two linear equations using the
  substitution method
• Recognize the connections between the three
  methods of solution
• Interpret the solution to a system of two linear
  equations in terms of the problems content

3
Vocabulary

• System of linear equations two equations that
  relate the same two variables
• Numerical method using a table of values to see
  with input results in the same output for the two
  equations
• Graphical method graph the functions and
  determine the coordinates of the point of
  intersection
• Substitution method an algebraic method using
  substitution to reduce the problem to one
  variable
• Consistent has exactly one solution (the graphs
  of the lines intersect)
• Inconsistent has no solutions (the graphs of
  the lines are parallel)

4
Solving a System of 2 Linear Equations

• Numerically by completing a table and noting
  which x-value gives you the same y-value
• Graphically by graphing the equations and
  finding their point of intersection
• Algebraically by using properties of equality
  to solve the equations for one variable and then
  the other
• Substitution method (also known as elimination)
• Addition method (the emphasis in lesson 3.3)

5
Business Checking Account

• In setting up your part-time business, you have
  two choices for a checking account at the local
  bank.
• If you anticipate making about 50 transactions
  each month, which checking account will be more
  economical?

MONTHLY FEE TRANSACTION FEE
REGULAR 11.00 0.17 for each transaction
BASIC 8.50 0.22 for each transaction in excess of 20
Basic
6
Business Checking Account
MONTHLY FEE TRANSACTION FEE
REGULAR 11.00 0.17 for each transaction
BASIC 8.50 0.22 for each transaction in excess of 20

• Let x represent the number of transactions.
  Write an equation that expresses the total
  monthly cost, C, for the regular account.
•  
•  
•  
• 3. Let x represent the number of transactions.
  Write an equation that expresses the total
  monthly cost, C, for the basic account. This is
  a more complicated equation because the
  transaction fee does not apply to the first
  twenty checks.

C 11 0.17t
C 8.50
for t 20 C 8.50 0.22(t
20) 0.22t 4.10 for t gt 20
7
Fill in the Table

• Use the two equations from the previous slideC
  11 0.17x and C 4.10
  0.22x

Number of Transactions 20 50 100 150 200 250 300
Cost of Regular ()
Cost of Basic ()
14.40 19.50 28.00 36.50 45.00 53.50
62.00
8.50 15.10 26.10 37.10 48.10 59.10
70.10
8
TI-83 Table Feature

• Use the table feature of your calculator to
  determine the x-value that produces two identical
  y-values
• TABLE (2nd Graph)
• Put in x values of interest (independent) in
  table
• Set Y1 to equation of interest
• Go back to table to read off dependent values

9
TI-83 Graph

• Use your calculator to graph the functions

window Xmin 0 Xmax 300 Xscl 10 Ymin 0 Ymax
80 Yscl 10 Xres 1
10
Graphical Method

• Step 1 Solve both equations for y
• Step 2 Put into your calculator (y1 for one
  and y2 for the other) and graph (or graph by
  hand)
• Step 3 If the lines intersect, then the
  intersection point is the solution if the lines
  are parallel, then there is no solution and if
  the lines are the same, then there are an
  infinite number of solutions
• Step 4 Write the solution (intersection point)
  (use TRACE on your calculator to find it)

11
Graphical Method - Solutions

• Consistent
  InconsistentOne Solution No
  Solution ? Solutions

12
Substitution Method

• Step 1 Solve one or both equations for a
  variable (both x or both y )
• Step 2 Substitute the expression that
  represents the variable in one equation for that
  variable in the other equation
• Step 3 Solve the resulting equation for the
  remaining variable
• Step 4 Substitute the value from step 3 into
  one of the original equations and solve for the
  other variable

13
Substitution Example

• Given y x 9 and y 3x 3
• Step 1 y 9 x and y 3x 3
• Step 2 9 x 3x 3
• Step 3 x x 9 4x
  3 3 3
  12 4x 3 x
• Step 4 y 3 9 or y 3(3)
  3 y 6 or y
  9 3 6

14
Problem 1

• Given y 3x 10 and y 5x 14
  Solve using substitution
• Step 1 y 3x 10 and y 5x 14
• Step 2 3x 10 5x 14
• Step 3 10 10
  3x 5x 24 - 3x - 3x
  0 2x 24
  -24 2x -12 x
• Step 4 y 3(-12) 10 -46

15
Problem 2

• Use the substitution method to sol the following
  system of checking account cost functions C
  0.17x 11 and C 0.22x
  4.10
• Step 1 C 0.17x 11 and y 0.22x 4.10
  
• Step 2 0.17x 11 0.22x 4.10
• Step 3 -.17x -.17x
  11 0.05x 4.10
  - 4.1 - 4.1
  6.9 0.05x 138
  x
• Step 4 0.17(138) 11 23.46 11 34.46

16
Summary and Homework

• Summary
• The solution of a system of equations is the set
  of all ordered pairs that satisfy both equations.
• The three standard methods for solving a system
  of equations are the Numerical method, Graphical
  method and Substitution method.
• A linear system is consistent if there is at
  least one solution, the point of intersection of
  the graphs.
• A linear system is inconsistent if there is no
  solution -- that is, the lines are parallel.
• Homework
• 1- 4, 7, 8