2'1 Linear Equations in Two Variables, p' 174183

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2.1 Linear Equations in Two Variables, p. 174-183

• OBJECTIVES
• Find slopes of lines
• Write linear equations in two variables

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• Slope-Intercept Form of the Equation of a Line
• p. 174
• The graph of the equation
• y mx b
• is a line whose slope is m and y-intercept is (0,
  b).

m units mgt0 Slope is positive
y-intercept (0,b)
1 unit
3
1 unit
m units m lt 0 Slope is negative
(0,b) y-intercept

• The slope of a line can be interpreted as
• -a ratio if the x and y-axis have the same units.
• -a rate or rate of change if the x and y-axis
  have different units.

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Slope
y
(1, 2)
y rises 6 units
2 (-4)
x
(3, 4)
(1, 4)
x runs 4 units
1 (-3)
Steepness of line is
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Slope is the ratio of vertical rise to the
corresponding horizontal run, or the steepness of
the line.
y
(x2, y2)
y2 y1
x
(x1, y1)
(x2, y1)
Steepness is
Slope
x2 x1
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Slope p.176

• A line going through two points (x1, y1) and (x2,
  y2), where x1? x2, has slope m, where

A line that rises from left to right has a
positive slope. A line that falls from left to
right has a negative slope.
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p. 183 22 Plot the points and find the slope of
the line passing through the pair of points.(2,
4), (4, 4)
2 units
y
(2, 4)
8 units
x
Therefore the slope is 4. What is the equation
of the line?
(4, 4)
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m 4 y mx b y 4x b
y
(2, 4)
4 4 (2) b
4 8 b
x
12 b
y 4 x 12
(4, 4)
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How do we find the equation of the line through
the points (1, 2), ( 1, 4) ?
y
(1, 2)
Slope of line is
x
y 3/2x
b
(3, 4)
2 3/2(1) b
2 3/2 b
1/2 b
y 3/2x 1/2
Note the y-intercept is (0, 1/2)
This is called the slope-intercept form of the
line.
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Point-Slope Form of the Equation of a Line p.
178 The equation of the line with slope m
passing Through the point (x1, y1) is y y1
m( x x1 ).

• y y1 m( x x1 ) and replacing m with 4,
• y y1 4( x x1 ).
• Pick either of the two original points from which
  we found the slope, (2, 4) or (4, 4), and
  substitute in for (x1, y1).
• y 4 4( x 2 )
• y 4 4x 8
• y 4x 12

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• What if the slope is neither positive or negative?

y-intercept (0,b)
m 0 Zero implies a horizontal line y b
m is undefined Division by zero implies a
vertical line x a
x-intercept (a, 0)
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Parallel and Perpendicular Lines p. 179

• Parallel-Two distinct nonvertical lines with
  slopes that are equal. ( m1 m2 )
• Perpendicular-Two nonvertical lines with slopes
  that are negative reciprocals of each other. (
  m1 -1/m2 )

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p. 184 70

• Find equations of parallel and perpendicular
  lines to the line x y 7 and through the
  point (-3,2).
• What is the slope intercept form of x y 7 ?
• (General form)
• Slope intercept form y -x 7
• m1 -1 b 7
• slope of parallel line -1.
• Parallel line y 2 -1 ( x (-3) )
• y 2 -x 3
• y -x 1

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• Perpendicular lines have slopes that are negative
  reciprocals of each other.
• m2 slope of perpendicular line.
• m1 -1, then m2 1
• y 2 1( x (-3) )
• y 2 x 3
• y x 5

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Application

• A company has fixed costs of 12,000 and variable
  costs of 7.00 per unit manufactured. What is
  the monthly costs?
• x variable costs
• y fixed costs

VERBAL MODEL ALGEBRAIC y 7.00x
12,000 MODEL
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The company has 85,000 available. How many
units can be produced?

• y 7.00x 12,000
• y 85,000
• 85,000 7.00x 12,000
• 73,000 7.00x
• x 73,000/7.00 10,428.5714286.
• Thus, we may produce 10,428 units for 85,000.
• What is the rate of change per unit?
• The variable cost is 7.00 per unit.
• What is the y-intercept?
• (0, 12,000)

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Equations of Lines p. 182

• General form Ax By C 0
• Vertical line x a
• Horizontal line y b
• Slope-intercept form y mx b
• Point-slope form y y1 m(x x1)
• Two-point form

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Homework

• P. 183 1-84, 89-118 alternate odd
• Read p. 144-149 (1.7)
• PRE QUIZ 5
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