4'1: Systems of Linear Eqns

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4.1 Systems of Linear Eqns

• Basic Concepts
• Total of 184 million adults took a family
  vacation in 1998 or 1999
• 2 million more adults took a trip in 1999 than in
  1998
• Expressed mathematically
• X y 184 the total was 184
• X y 2 the diff. Was 2

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A system of linear eqns in 2 variables

• x y 184
• x y 2
• What does x represent?
• What does y represent?
• A system of linear eqns in 2 variables may or may
  not have a unique solution (x, y)

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Testing for a solution

• (0, 3) or (-1, 2)
• Which is a solution to the system
• -x 4y 9
• 3x 3y -9
• We determine this by substituting the values of x
  and y into the system. To be a solution we must
  have TRUE statements for both equations.

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Determine a solution!

• (-3, -1) or (3, 1) for
• X 3y 0
• 3x y 10
• (4, 0) or (3, 5) for
• 5x 4y 20
• -x 4y -4

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Solving Graphically

• x y 184
• x - y 2
• To solve by sketching a graph we must be
  extremely accurate!

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Sketch graph ofx y 184x - y 2
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Solve graphically on the TI-83

• x y 184
• x - y 2
• We must first solve each eqn for y
• Y 184 - x
• Y x - 2
• Enter both eqns in the Y screen
• GRAPH

8
Adjusting the Window

• You can try ZOOM - 0ZoomFit
• The calculator doesnt always know what you want
  to see.
• Adjusting the window manually
• WINDOW and adjust x-min, x-max, y-min, y-max
  until you can see the intersection of the 2
  graphs.

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Finding the point of Intersection

• CALC 2nd - TRACE
• 5intersect
• First Curve? (cursor should be on Y1) press
  ENTER.
• Second Curve? (cursor should be on Y2) press
  ENTER
• Guess? . Move cursor close to intersection point
  ENTER

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Finding the point of Intersection

• We are told that the intersection of the 2 graphs
  occurs at x 93, and y 91
• We can write the solution (93, 91)
• Interpret the solution in the context of our
  original problem
• (s. 1 .. Return to s.9)

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Now You!

• X - 3y 0
• 3x y 10

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Now You!

• Turn off, but DO NOT CLEAR Y1 and Y2 enter the
  next 2 eqns in Y3 and Y4.
• 5x - 4y 20
• -x 4y - 4
• Follow previous procedure, only now First curve
  Y3, and Second Curve Y4

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Solving a system numerically

• X - 3y 0
• 3x y 10
• We already have these eqns in our calculator
  Turn on Y1 and Y2 turn off Y3 and Y4

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Solving a system numerically

• TBLSET 2nd - Window
• Select a suitable starting point and
  delta-Table
• Table 2nd - Graph
• Look in Y1 and Y2 columns to find equal values
  this is where the eqns are equal!
• We find these at x 3, and y 1

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Solving a system numerically

• 5x - 4y 20
• -x 4y - 4
• Turn off Y1 Y2 Turn on Y3 Y4 and follow
  same procedure to find solution in table.
• Change delta-Table to

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Types of Linear Systems

• Consistent
• At least 1 solution
• Graph 2 intersecting lines
• Or
• 2 Identical lines called a Dependent System
• X 2y 3
• 2x 4y 6

• Inconsistent
• No solutions!
• Graph 2 parallel lines
• X 2y 3
• 2x 4y ?

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Recognizing an Inconsistent system

• 4x 2y 44
• 2x y 20

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Recognizing an Inconsistent system

• 3x - y 7
• 9x - 3y ?
• What could I use for ? to make the system
  Inconsistent (parallel lines, .. No solutions)
• What could I use for ? to make the system
  Dependent (lines coincide, an infinite number
  of solutions)

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Writing solving Systems

• The sum of 2 numbers is 18, and their difference
  is 6. Find the 2 numbers.
• X y 18
• X - y 6

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Writing solving Systems

• Twice a number minus a second number is 5 The
  sum of the two numbers is 16. Find the 2 nos.
• 2x - y 5
• X y 16

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Writing solving Systems

• A car was driven for 2 hours, part of the time at
  40 mph and the rest of the time at 60 mph. The
  total distance traveled was 90 miles. How long
  did the car travel at each speed?
• X y 2
• 40x 60y 90

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Writing solving Systems

• In 1998, Mark McGwire and Sammy Sosa hit a total
  of 136 home runs. Mark hit 4 more home runs than
  Sammy. How many hrs did each player hit?
• X y 136
• X 4 y

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HW Sec. 4.1

• 1,3,5 7,9 11-17 21-29 odds 33,35 43,44
  53-56.