Applied Calculus Math 215 - University of Hawaii

12 CHAPTER 1. SOME BACKGROUND MATERIAL
Example 1.6. The lines
2x +5y=7 & 4x+10y=15
are parallel and have no intersection point.
To see this, observe that the first equation, multiplied with 2, is 4x +
10y = 14. There are no numbers x and y for which 4x +10y =14and
4x+10y= 15 at the same time. Thus this system of two equations in two
unknowns has no solution, and the two lines do not intersect. ♦
To be parallel also means to have the same slope. If the lines are not
vertical (b = 0andB= 0), then the condition says that the slopes −a/b of
the line l1 and −A/B of the line l2 are the same. If both lines are vertical,
then we have not assigned a slope to them.
If Ab = aB, then the lines are not parallel to each other, and one can
show that they intersect in exactly one point. You saw an example above.
If Aa = −bB, then the lines intersect perpendicularly. Assuming that
neither line is vertical (b = 0andB= 0), the equation may be written as
a A
×
b B
= −1.
This means that the product of the slopes of the lines (−a/b is the slope
ofthefirstlineand−A/B the one of the second line) is −1. The slope
of one line is the negative reciprocal of the slope of the other line. This is
the condition which you have probably seen before for two lines intersecting
perpendicularly.
Example 1.7. The lines
3x − y =1 & x+3y=7
have slopes 3 and −1/3, resp., and intersect perpendicularly in (x, y) =
(1, 2). ♦
Exercise 6. Find the intersection points of the lines
l1(x) =3x+4 & l2(x)=4x−5.
Sketch the lines and verify your calculation of the intersection point.