Lines with Opposite-Signed Slopes
Take two lines, each with opposite signed slopes. One slope has a positive slope, the other has a negative slope. In general, the pair of lines will always intersect at one point.
Define two lines as such:
y = m1 ∙ x + b1, where m is the slope and b is the y-intercept.
y = m2 ∙ x + b2
Assume that m1 and m2 are not zero, and m1 > 0 and m2 < 0.
If m2 < 0, -|m2| < 0 (see Aside)
and as a result, m2 = -|m2|
Also, since m1 > 0, |m1| > 0, and m1 = |m1|.
Equating both lines and solving for x:
m1 ∙ x + b1 = m2 ∙ x + b2
|m1| ∙ x + b1 = -|m2| ∙ x + b2
|m1| ∙ x + |m2| ∙ x = b2 - b1
x = (b2 - b1) ÷ (|m1| + |m2|)
Since m1 and m2 are not zero, the above solution is defined.
QED
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Aside: If x < 0, then -|x| < 0
Assume x is not zero.
By definition, the absolute value of x, denoted as |x|, is the defined as the distance x is from 0 and is always positive.
Then:
|x| > 0
Multiply both sides by -1:
-|x| < 0
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Eddie
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