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Online Volume 2, Three Skills for Algebra, Chapters 1 to 25 - skip 18., verbalizes and explains key skills and concepts, those needed in calculus, again to make the hard easier. A visual understanding of complex numbers may serve as back ground info for partial fraction decomposition. |
Deriving the point slope Equation for Straight Lines - Four Cases
In this lesson, we explain how equations for lines in the coordinate plane follow from similarity of triangles in the plane..
We will see or show below that points (x,y) on non-vertical line L satisfy point-slope equation
y - y1 = m(x-x1)
where m the slope is a proportionality constant, a real number, and (x1, y1) is a given point on the line. The equation of vertical line through a point (x1, y1) is
x = x1
The derivation of the above two equations for four cases
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Downward Slanted Line: y - y1 = m(x-x1)
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Upward Slanted Line: y - y1 = m(x-x1)
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Horizontal Line: y - y1 = 0
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Vertical Line: x - x1= 0
follow below.
Downward Slanted Lines - Case 1 of 4
For a downward slanting line, every pair of triangles with hypotenuse on the
line and sides parallel to the coordinate axes are similar. Similarity
follows because the line through the hypotenuses serves as a transversal for the
parallel lines through the horizontal and vertical sides. From latter
corresponding angles are equal.
Two triangles one with sides lengths given by lower case and another with lengths in uppercase characters.
Similarity implies corresponding sides are proportional. That is for some constant q
drop = q DROP and run = q RUN
The latter gives
| drop run |
= | q*DROP q*RUN |
= | DROP RUN |
Hence the drop over run ratios for the triangles are equal.
| drop run |
= | DROP RUN |
The common value of these ratios provides a proportionality constant k such that the drop in each triangle is proportional to the run of each triangle.
Calculation of the drop over run proportionality constant.
The proportionality constant k can be computed from any pair of points (x1,
y1) and (x2, y2) are the line.
From the diagram, the drop over rise proportionality constant is
| k | = | DROP RUN |
= |  |
That is drop over rise proportionality constant k is given by the calculation
| k | = | |
Equation that follows from proportionality
Now if (x, y) is on the line with drop over run proportionality constant k and (x1, y1) is a point on the with x1 < x on the line we have a situation like the following.
Diagram for x1 < x
For the run and drop triangle determine by the two points (x,y) and (x1, y1), the proportionality constant
| k | = |  |
Therefore (x, y) satisfies the equation
y1 - y = k(x-x1)
or equivalently
y - y1 = -k(x-x1)
where (x1, y1) and k are unknown. The change of notation m = -k puts this equation in the sought after form
y - y1 = m(x-x1)
The value of m is called the slope of the line. Here m = -k is negative as the drop over run ratio or proportionality constant k is positive - a situation implied by non-zero drop. Positive slopes will be seen below in the case of upward slanting lines.
Exercise: Show the same equation follows if (x, y) and (x1, y1) are two points on a line with proportionality constant k = -m with x > x1
Diagram for x1 > x
Lines Slanted Upward - Case 2 of 4
For a upward slanting line, every pair of triangles with hypotenuse on the line and sides parallel to the coordinate axes are similar. Similarity follows because the line through the hypotenuses serves as a transversal for the parallel lines through the horizontal and vertical sides. From latter corresponding angles are equal.
Similarity implies the rise over run ratios for the triangles are equal.
| rise run |
= | RISE RUN |
The common value of these ratios provides a proportionality constant K such that the rise in each triangle is proportional to the run of each triangle.
Calculation of the Proportionality Constant
The proportionality constant K can be computed from any pair of points (x1, y1) and (x2, y2) are the line.
Therefore
Equation that follows from proportionality
The assumption that (x,y) is on the line in place of (x ,y)
leads to the equation
where the proportionality constant K is known. The latter equation in turn implies
y - y1 = K(x-x1)
The latter gives an equation satisfied by all points on the line through point (x1, y1) with rise over run proportionality constant K. The point-slope equation for a line
y - y1 = m(x-x1)
appears if we let m = K.
Horizontal Line - Case 3 of 4
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A horizontal line in the coordinate plane is parallel to the x-axis.
All points on the line have the same y-coordinate while the x-coordinate can have any value. If (x1, y1) is a point on a horizontal line then another point (x,y) is on the line when and only when y = y1 . This equation restricts y to the constant value y1 with out restricting the value of x. So the equation y = y1 .or equivalently y - y1 = 0 describes the horizontal line through (x1, y1). The latter equation y - y1 = 0 has the form y - y1 = m(x-x1) if we put m = 0 as zero time (x-x1) gives 0 regardless of the values of x and x1. |
Vertical Line - Case 4 of 4.
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A vertical line in the coordinate plane is parallel to the y-axis.
x = x1 describes the vertical line through (x1, y1). We are done.
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All points on
the line have the same x-coordinate while the y-coordinate can have any
value. If (x1, y1) is a point on a
vertical line then another point (x,y) is on the line when and only
when x = x1 . This equation restricts x to the
value x1 with out restricting the value of
y. So the equation Conclusion
For each non-vertical slanted line through a point (x1, y1), there is a constant m called the slope of the line such that every point (x,y) on the line satisfies the equation
y - y1 = m(x-x1)
Note if we call x-coordinate difference (x-x1) a run and vertical line difference y - y1 a rise, differences that may be positive, zero or negative, the point slope equation say the rise is proportional to the run with proportionality constant m.
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For each straight line, its slope m is the proportionality constant linking the rises to runs for any two points on the line.
Two points are usually needed to compute the slope. For a straight line segment, the slope m is a constant of proportionality between Dy = y-y1 and Dx = x-x1. The change Dy in y (the rise) is proportional to the change in Dx (the run) in x, and so
or equivalently
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For each point on the line (x1, y1), there is a point-slope equation. But given (x1, y1), we can take the equation y - y1 = m(x-x1) and express y in terms of x. That operation yields y = mx +b where b = y1- mx1. Now x = 0 in y = mx +b implies y = b. So the point (0,b) gives the lines intersetion with the y-axis. Since the line is not vertical, it has only one point of intersection. So b b = y1- mx1 is unique and independent of the choice of (x1, y1). The convention that we rewrite the point-slope form y - y1 = m(x-x1) as y = mx+b by solving for y provides the unique y-intercept slope equation of the line L. |
Remark: If we know points (x1, y1) and (x2, y2) belong to non-vertical line with equation
y - y1 = m(x-x1)
then (x,y) = (x2, y2) must satisfy the equation. So
y2 - y1 = m(x2-x1)
Thus the value of m, known or not, must be given by the equation
where the RISE = y2 - y1 and RUN = x2-x1 are allowed to be real numbers. The slope m provides the (signed) rise over (signed) run proportionality constant or ratio for the line.
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