Linear Systems of Equations

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CHAPTER 6

• Linear Systems of Equations

2
SECTION 6-1

• Slope of a Line and Slope-Intercept Form

3

• COORDINATE PLANE consists of two perpendicular
  number lines, dividing the plane into four
  regions called quadrants

4

• X-AXIS - the horizontal number line
• Y-AXIS - the vertical number line
• ORIGIN - the point where the
• x-axis and y-axis cross

5

• ORDERED PAIR - a unique assignment of real
  numbers to a point in the coordinate plane
  consisting of one x-coordinate and one
  y-coordinate
• (-3, 5), (2,4), (6,0), (0,-3)

6
COORDINATE PLANE
7

• LINEAR EQUATION
• is an equation whose graph is a straight line.

8

• SLOPE
• is the ratio of vertical change to the
  horizontal change. The variable m is used to
  represent slope.

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FORMULA FOR SLOPE

• m change in y-coordinate
• change in x-coordinate
• Or
• m rise
• run

10

• SLOPE OF A LINE
• m y2 y1
• x2 x1

11

• Find the slope of the line that contains the
  given points.
• M(4, -6) and N(-2, 3)
• M(-2,1) and N(4, -2)
• M(0, 0) and N(5, 5)

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• Y-Intercept
• is the point where the line intersects the y
  -axis.

13

• X-Intercept
• is the point where the line intersects the
• x -axis.

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• HORIZONTAL LINE
• a horizontal line containing the point
• (a, b) is described by the equation y b

15

• VERTICAL LINE
• a vertical line containing the point (c, d) is
  described by the equation x c

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• SLOPE-INTERCEPT FORM
• y mx b
• where m is the slope and b is the y -intercept

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• Find the Slope and Intercept
• y 2x - 7
• 2y 4x 8
• 2x 2y 4
• -4x 7y 28

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SECTION 6-2

• Parallel and Perpendicular Lines

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• SLOPE of PARALLEL LINES
• Two lines are parallel if their slopes are equal

20

• Find the slope of a line parallel to the line
  containing points M and N.
• M(-2, 5) and N(0, -1)

21

• Find the slope of a line parallel to the line
  containing points M and N.
• M(3, 5) and N(0, 6)

22

• Find the slope of a line parallel to the line
  containing points M and N.
• M(-2, -6) and N(2, 1)

23

• SLOPE of PERPENDICULAR LINES
• Two lines are perpendicular if the product of
  their slopes is -1

24

• Find the slope of a line perpendicular to the
  line containing points M and N.
• M(4, -1) and N(-5, -2)

25

• Find the slope of a line perpendicular to the
  line containing points M and N.
• M(3, 5) and N(0, 6)

26

• Find the slope of a line perpendicular to the
  line containing points M and N.
• M(-2, -6) and N(2, 1)

27

• Determine whether each pair of lines is parallel,
  perpendicular, or neither
• 7x 2y 14
• 7y 2x - 5

28

• Determine whether each pair of lines is parallel,
  perpendicular, or neither
• -5x 3y 2
• 3x 5y 15

29

• Determine whether each pair of lines is parallel,
  perpendicular, or neither
• 2x 3y 6
• 8x 4y 4

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SECTION 6-3

• Write Equations for Lines

31

• POINT-SLOPE FORM
• y y1 m (x x1)
• where m is the slope and (x1 ,y1) is a point on
  the line.

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Write an equation of a line with the given slope
and through a given point

• m-2
• P(-1, 3)

33
Write an equation of a line through the given
points

• A(1, -3) B(3,2)

34
Write an equation of a line with the given point
and y-intercept

• b3 P(2, -1)

35
Write an equation of a line parallel to y-1/3x1
containing the point (1,1)

• m-1/3
• P(1, 1)

36
Write an equation of a line perpendicular to
y2x1 containing the point (2,1)

• M2
• P(2, 1)

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SECTION 6-4

• Systems of Equations

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• SYSTEM OF EQUATIONS
• Two linear equations with the same two variable
  form a system of equations.

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• SOLUTION
• The ordered pair that makes both equations true.
  

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• SOLUTION
• The point of intersection of the two lines.

41

• INDEPENDENT SYSTEM
• The graph of each equation intersects in one
  point.

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• INCONSISTENT SYSTEM
• The graphs of each equation do not intersect.

43

• DEPENDENT SYSTEM
• The graph of each equation is the same. The
  lines coincide and any point on the line is a
  solution.

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• SOLVE BY GRAPHING
• 4x 2y 8
• 3y -6x 12

45

• SOLVE BY GRAPHING
• y 1/2x 3
• 2y x - 2

46

• SOLVE BY GRAPHING
• x y 8
• x-y 4

47
SECTION 6-5

• Solve Systems by Substitution

48

• SYSTEM OF EQUATIONS
• Two linear equations with the same two variable
  form a system of equations.

49

• SOLUTION
• The ordered pair that makes both equations true.
  

50

• SOLUTION
• The point of intersection of the two lines.

51

• PRACTICE USING DISTRIBUTIVE LAW
• x 2(3x - 6) 2
• x 6x 12 2
• 7x -12 2
• 7x 14
• x 2

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• PRACTICE USING DISTRIBUTIVE LAW
• -(4x 2) 2(x 7)

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• SUBSTITUTION
• A method for solving a system of equations by
  solving for one variable in terms of the other
  variable.

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• SOLVE BY SUBSTITUTION
• 3x y 6
• x 2y 2
• Solve for y in terms of x.
• 3x y 6
• 3x 6 y
• 3x 6 y then

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• SOLVE BY SUBSTITUTION
• Substitute the value of y into the second
  equation
• x 2y 2
• x 2(3x 6) 2
• x 6x 12 2
• 7x 14
• x 2 now

56

• SOLVE BY SUBSTITUTION
• Substitute the value of x into the first equation
• 3x y 6
• y 3x 6
• y 3(2 6)
• y 3(-4)
• y -12

57

• SOLVE BY SUBSTITUTION
• 2x y 0
• x 5y -11
• Solve for y in terms of x.
• 2x y 0
• y -2x
• then

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• SOLVE BY SUBSTITUTION
• Substitute the value of y into the second
  equation
• x 5y -11
• x 5(-2x) -11
• x 10x -11
• 11x -11
• x -1

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• SOLVE BY SUBSTITUTION
• Substitute the value of x into the first equation
• 2x y 0
• y -2x
• y -2(-1)
• y 2

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SECTION 6-6

• Solve Systems by Adding and Multiplying

61

• ADDITION/SUBTRACTION METHOD
• Another method for solving a system of equations
  where one of the variables is eliminated by
  adding or subtracting the two equations.

62

• STEPS FOR ADDITION OR SUBTRACTION METHOD
• If the coefficients of one of the variables are
  opposites, add the equations to eliminate one of
  the variables. If the coefficients of one of the
  variables are the same, subtract the equations to
  eliminate one of the variables.

63

• STEPS FOR ADDITION OR SUBTRACTION METHOD
• Solve the resulting equation for the remaining
  variable.

64

• STEPS FOR ADDITION OR SUBTRACTION METHOD
• Substitute the value for the variable in one of
  the original equations and solve for the unknown
  variable.

65

• STEPS FOR ADDITION OR SUBTRACTION METHOD
• Check the solution in both of the original
  equations.

66

• MULTIPLICATION AND ADDITION METHOD
• This method combines the multiplication property
  of equations with the addition/subtraction
  method.

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• SOLVE BY ADDING AND MULTIPLYING
• 3x 4y 10
• 3y 2x 7

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• SOLUTION
• 3x 4y 10
• -2x 3y -7
• Multiply equation 1 by 2
• Multiply equation 2 by 3

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• SOLUTION
• 6x 8y 20
• -6x 9y -21
• Add the two equations.
• y -1

70

• SOLUTION
• Substitute the value of y into either equation
  and solve for
• 3x 4y 10
• 3x 4(-1) 10
• 3x 4 10
• 3x 6
• x 2

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SECTION 6-7

• Determinants Matrices

72

• MATRIX
• An array of numbers arranged in rows and columns.

73

• SQUARE MATRIX
• An array with the same number of rows and
  columns.

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• DETERMINANT
• Another method of solving a system of equations.

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• DETERMINANT OF A SYSTEM OF EQUATIONS
• The determinant of a system of equations is
  formed using the coefficient of the variables
  when the equations are written in standard from.

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• DETERMINANT VALUE
• Is the difference of the product of the diagonals
  (ad bc).
• a b
• c d

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• SOLVE USING DETERMINANTS
• x 3y 4
• -2x y -1

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• SOLVE USING DETERMINANTS
• x 3y 4
• -2x y -1
• Matrix A 1 3
• -2 1

79

• SOLVE USING DETERMINANTS
• Matrix Ax 4 3
• -1 1
• x det Ax /det A

80

• SOLVE USING DETERMINANTS
• det Ax 4(1) (3)(-1)
• 4 3
• 7

81

• SOLVE USING DETERMINANTS
• Det A 1(1) (3)(-2)
• 1 6
• 7 thus
• x 7/7 1

82

• SOLVE USING DETERMINANTS
• Matrix Ay 1 4
• -2 -1
• y det Ay /det A

83

• SOLVE USING DETERMINANTS
• det Ay -1(1) (4)(-2)
• -1 8
• 7 thus
• y 7/7 1

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SECTION 6-8

• Systems of Inequalities

85

• SYSTEM OF LINEAR INEQUALITIES
• A system of linear inequalities can be solved by
  graphing each equation and determining the region
  where the inequality is true.

86

• SYSTEM OF LINEAR INEQUALITIES
• The intersection of the graphs of the
  inequalities is the solution set.

87

• SOLVE BY GRAPHING THE INEQUALITIES
• x 2y lt 5
• 2x 3y 1

88

• SOLVE BY GRAPHING THE INEQUALITIES
• 4x - y ? 5
• 8x 5y 3

89
SECTION 6-9

• Linear Programming

90

• LINEAR PROGRAMMING
• A method used by business and government to help
  manage resources and time.

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• CONSTRAINTS
• Limits to available resources

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• FEASIBLE REGION
• The intersection of the graphs of a system of
  constraints.

93

• OBJECTIVE FUNCTION
• Used to determine how to maximize profit while
  minimizing cost

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• END