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Equations
of straight line
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Slope of a line |
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Slope-intercept form
of a line |
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The point-slope form
of a line |
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The two point form of
the equation of a line |
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Parallel and
perpendicular lines |
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Equations
of straight line |
| Definition of the
slope of a line |
| The slope
of a line is the ratio of its vertical change to its horizontal
change, or it is the tangent of the angle between the direction
of the line and the x-axis. |
| The slope
is the difference quotient |
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| Slope-intercept form
of a line |
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| The
point-slope form of a line
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| The
equation of a line that passes through the given point (x1,
y1)
and has the given slope m
is represented by the
definition of the slope and is called point-slope form or the
gradient form of the line. |
| Since the slope
of a line is the ratio of its vertical change to its horizontal
change then |
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 |
 |
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or y
-
y1 = m(x
-
x1) |
| The
equation can also be considered as the translation of
the source linear function y
= mx to the
point P1(x1,
y1). |
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| Example:
Find the equation of the line that is parallel with the line y
= - x
- 2 and passes through the point |
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P1( 2,
1) . |
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| The two point form of
the equation of a line |
| Two
points P1(x1,
y1)
and P2(x2,
y2)
determine a unique line on the Cartesian plane, therefore their
coordinates satisfy the equation y
= mx
+ c. |
| The
equation of the line which passes through the point P1(x1,
y1)
is y -
y1
= m(x
- x1).
As the point |
| P2(x2,
y2)
lies on the same line, its coordinates must satisfy the same
equation, so y2
- y1
= m(x2
- x1). |
| Thus,
the slope |
 |
| then |
 |
| is
the equation of the line passing through the two
points. |
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| Example:
Find the equation of the line which passes through points P(-2,
3) and Q(6,
-1). |
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| Parallel and perpendicular lines |
| Two
lines having slopes m1
and m2
are parallel if |
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m1
= m2 |
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that
is, if they have the same slope. |
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| To
acquire the criteria when two lines, y
= m1x
and y
= m2x |
| are
perpendicular or orthogonal we can
use the principle of similar triangles, OA'A
and OB'B
in the picture. |
| Therefore, |
m1
: 1 =
-1
: m2 |
=> |
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| This
relation will stay unchanged if we translate the perpendicular
lines, that is, when lines |
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y = m1x
+ c1
and y
= m2x
+ c2
are written in the slope-intercept form. |
| Two
lines are perpendicular if the slope of one line is the negative
reciprocal of the other. |
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| Example:
Find the equation of the line that is perpendicular to
the line |
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and passes
through |
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the point A(-2,
5). |
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| Pre-calculus contents
E |
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