Given two points , 
and
, the slope of a line is :
.
Given that m represents the slope and b represents the y-intercept, the
slope-intercept form of a
line is: . y=mx+b
Given that m represents the slope and is a
point, the point-slope form of a line is:
Example 1: Find the slope of the line between the points
(3,-5) and (-1, -2).
Solution 1: We use the slope formula with
to
get 
.
Practice 2: Find the slope of the line between the points (-1,7) and (4, 5).
Example 3: Find the slope and y- intercept of the line 3x-2y=4.
Solution 3: To see the slope and the y-intercept from the equation we need the
slope-intercept form of the
equation so we need to solve the equation for y. First subtract the x term from
both sides 3x-2y-3x=4-3x (and write
the x term before the constant number to make things easier for yourself) to get
-2y = -3x+4. Then divide each term
by the coefficient on the y to get 
. Simplify
to see 
. We then can see our slope which
is the coefficient of the x term,
. Then we
can see the y-intercept which is the constant number, b= -2.
Practice 4: Find the slope and y-intercept of the line 2x+5y = -7.
Example 5: Find the equation of the line passing through the point (-1/2, 4)
with slope -3.
Solution 5: Use the point-slope form of the line with
, and m=-3. We
get 
and then simplify
. This is an equation of a line in
point-slope
form and if there are no further instructions you can stop here. If the question
requests the equation of the line in
slope-intercept form you should simplify further to get y by itself.
Distributing gives 
and adding 4
to both sides gives 
. We then need a common
denominator to combine the like terms .
gives 
.
Practice 6: Find the equation of the line in slope-intercept form passing
through the point (3,
2/3) with slope -2.
Parallel lines have the same slope. The slopes of
perpendicular lines are opposite sign
reciprocals of each other.
Example 7: Find an equation of the line that is parallel to the line
and passes
through the point (1, 5). Write your final answer in slope-intercept form.
Solution 7: The first step is to find the slope of the given line,
. So we solve that equation
for y. Add 2x to both sides 
and divide by 4,
. So we see that the slope of this line is ½
.
Since we need a parallel slope our slope will be the same, m= ½. Notice that we
do not use the y-intercept from the
given line for anything. We take the new slope and the point given and put them
in the point-slope equation.
. Then we solve for y. Distributing gives
and adding 5 to both sides gives
. We then need a common denominator to
combine the like terms. 
gives
. This is our final answer. If you want to go
directly to the slope-intercept method and not use the point
slope there is an alternative method to solving the problem. After you find the
slope of m= ½, put this into the
slope-intercept form, 
. Then put the values
of the point given in for x and y, 
. Solve
for
b by first multiplying 
, subtract the
constant number over, 
and make a common
denominator,
, and combine like terms,
. Then use the slope-intercept form again
substituting in the values for
m and b, but not for x and y. 
. Some people
find this second method easier to understand while others
find it more time con suming .
Example 8: Find an equation of the line that is perpendicular to the line
and passes
through the point (-3, 5/2).
Solution 8: First find the slope of the given line by solving for y. Subtract x
from both sides to get ,
then divide by 3 to get
. So the slope is -1/3. Perpendicular lines
have opposite sign
reciprocals so our new slope is m=3. Use this slope and the point given in the
point-slope method to get
. Since the instructions didn’t specify a
specific form of the line we can stop here.
Example 9: Graph the line . 
Solution 9: There are many methods you
can use to graph lines. Here is one method that is
quick to do if the line is in slope-intercept form. Plot
a point on the y-axis at the y-intercept, b=1 for us.
So our first point is (0,1). Then use the slope as
rise/run. Since our slope doesn’t look like a fraction
we make it 2/1. Then we rise up if the slope is
positive or down if the slope is negative according
the number in the numerator and the we run to the
right according to the number in the denominator. So
starting from our y-intercept of 1 we rise up 2 and run
over to the right 1. This gives us another point at
(1,3). Then draw a line through the 2 points.
Here are some practice problems for you.
1. Find the slope and y−intercept of the line 3x−5y=20.
2. Determine whether the lines 7x+5y=8 and 14x−10y=3 are parallel,
perpendicular, or
neither.
3. Find an equation of the line that passes through the point (−3,5) and has
slope -2. Write
your final answer in slope-intercept form.
4. Find an equation of the line that passes through the points (6, 7) and (-5,
1). Write your
final answer in slope-intercept form.
5. Determine whether the lines 10x−6y=13 and 5x−3y=9 are parallel,
perpendicular, or
neither.
6. Find an equation of the line that is perpendicular to the line 3x+4y=9 and
passes
through the point (2, 5). Write your final answer in slope-intercept form.
7. Find the slope-intercept form of the line that has slope
and y−intercept (0, 7).
8. Find an equation of the line that passes through the point (-1, 7) and is
parallel to the line
that passes through (2, -5) and (-4, -3). Write your final answer in
slope-intercept form.
9. Determine whether the lines 2x−3y=12 and 6x+4y=5 are parallel, perpendicular,
or
neither.
10. Find an equation of the line that is parallel to the line 2x−11y=8 and
passes through the
point (-3, 6).
11. Find the equation of the line that passes through the point (8, -4) and has
slope 
.
12. Find an equation of the line that passes through the point (3, -7) and is
perpendicular to
the line that passes through (2, -5) and (-1, 2). Write your final answer in
slope-
intercept form.
13. Graph the line 
.
14. Graph the line 
15. Which of the following lines are perpendicular to 4x−7y=10?
