Applied Calculus Math 215 - University of Hawaii

8 CHAPTER 1. SOME BACKGROUND MATERIAL
Definition 1.1. A line consists of the points (x, y) in the x−y-plane which
satisfy the equation
(1.3)
ax + by = c
for some given real numbers a, b and c, where it is assumed that a and b are
not both zero.
If b = 0, then we can write the equation in the form x = c/a, andthis
means that the solutions of the equation form a vertical line. The value for
x is fixed, and there is no restriction on the value of y. Lines of this kind
cannot be obtained if the line is specified by an equation as in (1.2). The
line given by the equation 2x = 3 is shown as the solid line in Figure 1.2.
If a = 0, then we can write the equation in the form y = c/b, andthis
means that the solutions of the equation form a horizontal line, the value
for y is fixed, and there is no restriction on the value of x. The line given
by the equation 2y = 5 is shown as the dashed line in Figure 1.2.
If b = 0,thenax + by = c translates into y = − a c
b x + b , and the equation
describes a line with slope −a/b and y-intercept c/b.
-2 -1 1 2 x
-2
-4
-6
y
Figure 1.1: y(x) =2x−3
Exercise 1. Sketch the lines 5x =10and3y=5.
3
2
1
-1 1 2 3 x
-1
y
Figure 1.2: 2x =3&2y=5
Exercise 2. Sketch and determine the y-intercept and slope of the lines
3x +2y=6and2x−3y=8.