Math 141 Lecture Notes for Section 1.2
Section 1.2 -
1
Straight Lines
Slope of a Non-vertical Line: If (x1 , y1 ) and (x2 , y2 ) are any two distinct points on a non-vertical
line L, then the slope m of L is given by
m=
∆y
y2 − y1
=
∆x
x2 − x1
(1)
Where ∆y and ∆x stand for the change in the variables y and x respectively.
Example 1.2.1:
Consider the following line: To find the slope of the line we simply take the points (−2, 5) and (3, −4)
8 y
(-2,5)
6
4
2
x
−8
−6
−4
−2
−2
2
4
6
8
−4 (3,-4)
−6
−8
and put them into the equation for slope:
m=
5 − (−4)
9
9
=
=− .
−2 − 3
−5
5
Parallel Lines: Two distinct lines L1 and L2 are parallel if their slopes are equal.
Example 1.2.2:
Consider the line connecting (−2, 5) and (3, −4) from the example above for L1 which has slope of
− 95 . This line is parallel to the line L2 connecting (2, 1) and (−3, 10) since the slope of L2 is:
m2 =
10 − 1
9
9
=
=− .
−3 − 2
−5
5
Perpendicular Lines: Let L1 and L2 be lines with slopes m1 and m2 respectively. L1 is perpendicular
to L2 (written as L1 ⊥L2 ) if
1
m1 = −
.
(2)
m2
Math 141 Lecture Notes for Section 1.2
2
Example 1.2.3:
Suppose that we have a line L connecting (3, 5) and (1, 4). This line has slope
m=
5−4
1
= .
3−1
2
Therefore, any line perpendicular to L has slope −2.
Point-Slope Form of a Line: A line L containing a point (x1 , y1 ) with slope m can be expressed in
point-slope form with the following equation:
y − y1 = m(x − x1 ).
(3)
You should note that the point-slope form follows from the definition of a line itself. If (x1 , y1 ) is a
point on L, then any other point on L must satisfy
m=
y − y1
x − x1
Multiplying both sides of this equation by the value (x − x1 ) gives us
m · (x − x1 )
=
y − y1
· (x − x1 )
x − x1
= y − y1
which is precisely the point-slope form of a line.
Example 1.2.4:
Let L be the line with slope 3 containing the point (2, 1). Then we can write the point-slope form of
L as
y−1
= 3(x − 2)
= 3x − 6
x-Intercept of a Line: Let L be a line, then the x-intercept of L is the value a for which (a, 0) is a
point on L. All lines have x-intercepts except for horizontal lines that are not the line y = 0.
y-Intercept of a Line: Let L be a line, then the y-intercept of L is the value b for which (0, b) is a
point on L. All lines have y-intercepts except for vertical lines that are not the line x = 0.
Example 1.2.5:
We can easily find the y-intercept of the line L defined by
y − 1 = 3x − 6
by setting x to 0 and solving for y. In this case, we have
y − 1 = 3(0) − 6
= −6
y − 1 + 1 = −6 + 1
y = −5
Math 141 Lecture Notes for Section 1.2
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so the y-intercept of L is −5.
Similarly, for the x-intercept, we set y to 0 and solve for x.
(0) − 1 = 3x − 6
−1 + 6 = 3x − 6 + 6
5 = 3x
3
x=
5
and the x-intercept of L is
3
.
5
Slope-Intercept form of a Line: Let L be a line with slope m with y-intercept b. The slope-intercept
form of L is the equation
y = mx + b.
(4)
The slope-intercept form is a direct application of the point-slope form. Where point-slope uses any
point (x1 , y1 ) on the line, the slope-intercept form uses precisely the point (0, b) (because b is the
y-intercept, we know this point is on the line) and computes as:
y − b = m(x − 0).
Which, by adding b to both sides we get
y = mx + b.
Example 1.2.6:
Consider a line L connecting the point (5, 7) with y-intercept 2. Then we know (0, 2) is on the line so
we can compute the slope of L
7−2
5−0
5
= = 1.
5
m=
Now that we know the y-intercept and the slope, we simply need to substitute to get the slope-intercept
form of L,
y = x + 2.
General Form of a Line: The general form of a line L is
Ax + By + C = 0
(5)
where A, B, and C are all constants, with the rule that A and B cannot both be 0.
Vertical Lines: A line whose general form has B = 0 is a vertical line. These lines have equations that
look like:
x=c
(6)
Math 141 Lecture Notes for Section 1.2
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where c is a constant.
Horizontal Lines A line whose general form has A = 0 is a vertical line. These lines have equations
that look like:
y=d
(7)
where d is a constant.
Example 1.2.7:
Let’s look at the line with general form 3x + 4y − 5 = 0. Using our definition for the general form we
have that A = 3, B = 4, and C = −5. Rather than try and transform this equation to the point-slope
form or slope-intercept form, here are a few quick ways to get information out of the general form.
Slope: The slope of the line is the following ratio:
−A
B
which in this case is − 34 .
y-intercept: The y-intercept of the line is the ratio:
−C
B
which in this case is 54 .
x-intercept: The x-intercept of the line is the ratio:
−C
A
which in this case is 53 .
Suggested Homework Problems: 1,3,5,9,13,15,17,19,21,23,27,31,33,39,43,45,55,57,59.