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1.2 Straight Lines
• Slope
• Point-Slope Form
• Slope-Intercept Form
• General Form
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Slope of a Vertical Line
Let L denote the unique straight line that passes through
the two distinct points (x1, y1) and (x2, y2).
If x1 = x2, then L is a vertical line, and the slope is
undefined.
y
L
(x1, y1)
(x2, y2)
x
Slope of a Nonvertical Line
If  x1, y1  and  x2 , y2  are two distinct points on a nonvertical
line L, then the slope m of L is given by
y y2  y1
m 
x x2  x1
The number y is a measure of the vertical change in y,
x
and
is a measure of the horizontal change in x. The
slope m is a measure of the rate of change of y with respect
to x.
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Examples
Find the slope m of the line that goes through the points
(4, 1) and (6, -3).
Solution
Choose (x1, y1) to be (4, 1) and (x2, y2) to be (6, -3).
With x1 = 4, y1 = 1, x2 = 6, y2 = -3, we find
y  y 3  1 4
m 2 1 

 2
x2  x1 6  4
2
Parallel Lines and Perpendicular Lines
Two lines are parallel if and only if their slopes are equal or both
undefined
Two lines are perpendicular if and only if the product of their
slopes is –1. That is, one slope is the negative reciprocal of the
3
4
other slope (ex.
).
and 
4
3
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Equations of Lines
• Let L be a straight line parallel to the y-axis. Then the
vertical line L is described by the sole condition x = a.
• Next, suppose L is a nonvertical line so that it has a
well-defined slope m.
• Suppose (x1, y1) is a fixed point lying on L and (x, y) is a
variable point on L distinct from (x1, y1). Then the slope
is given by
y  y1
m
x  x1
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Point-Slope Form
An equation of a line that passes through the point
with slope m is given by:
 x1, y1 
y  y1  m  x  x1 
Ex. Find an equation of the line that passes through (3,1) and has
slope m = 4.
y  y1  m  x  x1 
y  1  4  x  3
y  1  4 x  12
4 x  y  11  0
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
x-intercept and y-intercept
A straight line L that is neither horizontal nor vertical cuts the
x-axis an the y-axis at points (a, 0) and (0, b) respectively.
The numbers a and b are called the x-intercept and yintercept, respectively.
Now let L be a line with slope m an y-intercept. With the
point given by (0, b) and slope m, we have
y  b  m( x  0)
y  mx  b
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Slope-Intercept Form
An equation of a line with slope m and y-intercept  0,b  is
given by:
y  mx  b
Ex. Find an equation of the line that passes through (0,-4)
and has slope m 
4
.
5
y  mx  b
4
y  x4
5
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Example
Determine the slope and y-intercept of the line whose equation is
5 x  4 y  9.
Solution:
Rewrite the given equation in the
Step 1.
slope-intercept form  4 y  5 x  9
5
9
 y  x
4
4
5
9
Step 2. We have m  and b   .
4
4
5
The slope of the line is
and the y -intercept
4
9

of the line is  0,   .
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
4

Example
Find an equation of the line that passes through (-2, 1) and is
perpendicular to the line y  2 x  7.
Solution:
Step 1.
Step 2.
Since the slope of the line y  2 x  7 is 2,
1
we have the slope of the perpendicular line is m   .
2
1
y  y1  m  x  x1   y  1    x  2 
2
1
1
 y 1   x 1  y   x
2
2
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
General Form
The general form of an equation of a line is given by:
Ax  By  C  0
Where A, B, and C are constants and A and B are not both zero.
*Note: An equation of a straight line is a linear equation
and every linear equation represents a straight line.
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Vertical Lines
Can be expressed in the
form x = a
y
x=3
x
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Horizontal Lines
Can be expressed in the
form y = b
y
y=2
x
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Ex. Find an equation of the line that passes through the point
(–2, 3) and is parallel to the y – axis.
y
y-axis
(–2, 3)
x
Vertical Line: x = –2
Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc.
Applied Example
Suppose an art object purchased for $50,000 is expected to
appreciate in value at a constant rate of $5000 per year for
the next 5 years.
• Write an equation predicting the value of the art object for
any given year.
• What will be its value 3 years after the purchase?
Applied Example 11, page 16
Solution
Let
x = time (in years) since the object was
purchased
y = value of object (in dollars)
Then, y = 50,000 when x = 0, so the y-intercept is b =
50,000.
Every year the value rises by 5000, so the slope is m =
5000.
Thus, the equation must be y = 5000x + 50,000.
After 3 years the value of the object will be $65,000:
y = 5000(3) + 50,000 = 65,000