College Algebra 9th txtbk

138 CHAPTER 2 GRAPHS
SOLUTIONS (A) Substitute m 2 and in y = mx b to obtain y 2 3 b 5
3x 5.
3
(B) The y intercept of y 3 4 x 1 is 1 and the slope is 4. If we start at the point (0, 1)
and move four units to the right (run), then the y coordinate of a point on the line
must move up three units (rise) to the point (4, 2). Drawing a line through these
two points produces the graph shown in Figure 7.
5
y
5
4
3
5
x
5
Z Figure 7
MATCHED PROBLEM 4
5
Write the slope–intercept form of the line with slope 4 and y intercept 2. Graph the
equation.
y
(x, y)
x
In Example 4 we found the equation of a line with a given slope and y intercept. It is
also possible to find the equation of a line passing through a given point with a given slope
or to find the equation of a line containing two given points.
Suppose a line has slope m and passes through the point (x 1 , y 1 ). If (x, y) is any other
point on the line (Fig. 8), then
(x 1 , y 1 )
(x, y 1 )
y y 1
x x 1
m
Z Figure 8
that is,
y y 1 m(x x 1 )
(3)
Because the point (x 1 , y 1 ) also satisfies equation (3), we can conclude that equation (3) is an
equation of a line with slope m that passes through (x 1 , y 1 ).
Z THEOREM 3 Point–Slope Form
An equation of the line with slope m that passes through (x 1 , y 1 ) is
y y 1 m(x x 1 )
which is called the point–slope form.
If we are given the coordinates of two points on a line, we can use the given coordinates
to find the slope and then use the point–slope form with either of the given points to
find the equation of the line.