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Line and Definition of Slope

      Slope of a line (steepness)
Consider a particle moving along a non vertical line segment from a point p₁( x₁,y₁ ) to a point p₁( x₁,y₁ ). The vertical change y₂ – y₁ is called the rise, and the horizontal change x₂ – x₁ the run.

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      DEFINITION 1.4.1
If P(x₁, y₁) and P(x₂, y₂) are points on a nonvertical line, then slope m of the line is defined by:

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It does not matter which point is called P₁ and which one is called P₂
      Slope of  P₁P₂

= (y₂ - y₁)/(x₂ - x₁)

= -(y₁ - y₂)/[-(x₁ - x₂)]

= (y₁ - y₂)/(x₁ - x₂) = Slope of P₁P₂

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Any two distinct points on a nonvertical line can be used to calculate the slope of the line. To measure the slope, we generally move from left to right when measuring the distance travelling horizontally.
Because of this, sometimes the concept of fall replaces thar of rise!

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      Example
In each part find the slope of the line through
          (A)     (6, 2) and (9, 8)
          (B)     (2, 9) and (4, 3)
          (C)    (-2, 7) and (5, 7)

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Solution:
We know that slope of line through two points P₁(x₁, y₁) and p₁(x₁, y₁) , is given by
m= (y₂ - y₁)/ (x₂ - x₁)
So
    a) m= (8 - 2)/(9 - 6) = 6/3 = 2
On coordinate plane xy

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Similarly
    b) m= (3 - 9)/(4 - 2) = -6/2 = -3
On coordinate plane xy

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Also
    c) m= (7 -7)/[5 - (-2)] = 0/7 = 0
On coordinate plane xy

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      1.4.2 Definition (Angle of inclination)
For a line L is not parallet to x-axis, the angle of inclination is the smallest angle φ massured counterclockwise from the direction of the positive x-axis to L.
For a line parallel to the x-axis, we take φ = 0
As shown in the following figures.
       

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If m is slope of line then,
m = rise/run
    = Rate of change of y with respect to x

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      THEOREM 1.4.3
For a nonvertical line, the slope m and the angle of inclination φ are related by
            m = tan φ

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      Example:
Find the angle of inclination for a line of slope m = 1 and alsp for a line of slope m = -1

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      Solution:
  If m=1 tan φ = 1, so that φ = π/4 = 45°

  If m=-1 tan φ = -1 since, o < φ < π φ = 3π/4 = 135°

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      THEOREM 1.4.4 Let L₁ and L₂ be lines with slopes m₁ and m₂, respectively
  (a)   The lines are parallel if and only if      m₁ = m₂
  (b)   The lines are prependicular if and only if      m₁m₂ = -1

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      Proof: (a)
If L₁ and L₂ are non vertical lines, then their angles of inclination φ₁ and φ₂ are equal.
            φ₁ =φ₂
Thus
    m₁ = tanφ₁ = tanφ₂ = m₂

Conversely, if two slope lines are equal, I.e.
        M₁ = M₂
⇒     tan(φ₁) = tan(φ₂)
⇒         φ₁ = φ₂
So, lines are parallel.

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(b) Assume that φ₁ < φ₂
Then referring to the figure
m₁ = tanφ₁ = c/h

m₂ = tanφ₂ = -h/c

The proof of the converse is left as an exercise.

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      THEOREM 1.4.5
The vertical line through (a, 0) and the horizontal line throudgh (0, b) are represented, respectively, by the equation
x = a and y = b

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      THEOREM 1.4.6
The line Passing throudh P₁(x₁, y₁) and having slope m is given be the equation
            y - y₁ = m(x - x₁)
This is called the point-slope form of the line.

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      THEOREM 1.4.7
The line with y-intercept b and slope m is given by the equation
            y = mx + b
This is called the slope-intercept form of the line.