Gradient and Intercept

1 Key excerpts on "Gradient and Intercept"

• 

  eBook - ePub

  A First Course in Geometry

  • Edward T Walsh(Author)
  • 2014(Publication Date)
  • Dover Publications

    (Publisher)

  k), parallelism and perpendicularity are quite recognizable from the equations alone. In the more general case of nonvertical lines, the situation is almost as easy to deal with.

       Consider, as in the figure, any nonvertical line, intersecting the x- and y-axes at A (a, 0) and B(0, b) respectively. These points are called the x-intercept and the y-intercept. The point-slope form of the equation for this line is

                   y − b = m (x − 0)

  where m is the slope and B(0, b) is the point used. This equation can be rewritten in the form

                   y = mx + b

  Since m is the slope and b is the y-coordinate of the y-intercept, this form of a linear equation is called the slope-intercept form.

       Example 1 The equation of the line with slope 3 and y-intercept B(0, −2) is

                       y = 3x + 2

       Example 2 We can rewrite

                       3x − 2y + 7 = 0

       in slope-intercept form by solving for y as follows:

       Slope-intercept form is useful in establishing the following important result.

       THEOREM 119 Two distinct nonvertical lines are parallel iff they have the same slope.

       Proof: Suppose

                   y = m1 x + b1 is the equation of 1

                   y = m2 x + b2 is the equation of

  Then we have two things to prove :

           (i)  If m1 = m2 , then ||

           (ii)  If || , then m1 = m2

       The proof of (i) is indirect, as follows. Suppose m1 = m2 and . Then and must intersect at some point, the coordinates of which satisfy the equation of both lines. Therefore, for that point (x, y), we can write

                   y = m1 x + b1 and y = m1 x + b2

  or

                   b1 − y − m1 x = b2

  This cannot be, however, for in such a case the equations of the two lines would be identical and we would be considering only one line, not two. →/← Therefore, if m1 = m2 , then || .

       The proof of (ii) is as follows. Suppose || . If we perform the construction of the right triangles indicated in the figure, it is obvious that

                   ΔABC ~ ΔDEF

  and

  Therefore, if || , then m1 = m2 .

       Example 3 Suppose the equation of is 3x + 2y = 7; and suppose || and P(2, − 3) ∈ . We can find the equation of by noting that the equation of may be written

       so that the slope of is . Therefore, in point-slope form, the equation of

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