SLOPE OF A LINEAR EQUATION, Ax + By = C
The slope of a line is a ratio of the vertical change to the horizontal change between any two
points on a line.
Slope is denoted as "m".
Slope can be referred to in the following ways given below.
= change in y = Δ y = rise
change in x
Δx
run
slope = vertical change
horizontal change
FINDING SLOPE WHEN GIVEN TWO POINTS, (x1, y1) and (x2, y2):
The slope, m, of any two points, (x1, y1) and (x2, y2), on a line is defined by
slope = m =
y 2  y1
x2  x1
or
difference in y
difference in x
EXAMPLES:
1) Find the slope of a line that passes through the points (5, 9) and (-2, 4).
m = 9 - 4 = 5 is the slope.
5 - (-2)
7
Note: if you set up the formula as such,
2)
4-9 = -5
-2 - 5
-7
= 5 , the slope is still 5.
7
7
 1 
 2
Find the slope of a line that passes through the points,   ,0  and  2, 
 3 
 5
2
5
m =
=
7
1

 2
3
3
0
2
5

=
6
is the solution .
35
Note: Complex fractions should be reduce to simple fractions for the final slope solution
To simplify: numerator ÷ denominator
3) Find the slope of a line that passes through the points (3, 7) and (3, -10).
m = 7 - (-10) =
3-3
17 .
0
The slope is undefined.
NOTE: When the slope is undefined, then the graph of the line that passes through those
points is a vertical line.
4) Find the slope of a line that passes through the points (5, -4) and (6, -4).
m = -4 - (-4) = 0 = 0. The slope is zero.
5-6
-1
NOTE: When the slope is zero, then the graph of the line that passes through those points is a
horizontal line.
FINDING SLOPE FROM THE EQUATION OF A LINE:
Definition: y = mx + b is the slope-intercept form of the linear equation.
where m is the slope. (m is the coefficient of x)
b is the y-intercept, (0, b). (b is the constant)
(x, y) are the ordered pair values.
STEPS:
1) Convert the equation into slope-intercept form, if necessary, by solving for
"y" completely.
2) Identify the slope. (m = the coefficient of x)
EXAMPLES:
1) Find the slope in the linear equation, y = 3x - 5.
Since the equation of the line is in slope-intercept form, then m = 3.
2) Find the slope in the linear equation, 5x - 12y = 10.
Solve for y:
5x – 12y = 10
– 12y = 10 – 5x
5 5
y  x
6 12
Identify the slope:
is in slope-intercept form.
The slope is
5
.
12
3) Find the slope in the linear equation, 2(x - 4) - (3 - y) = 4x + 5y - 7.
Solve for y:
2x – 8 – 3 + y = 4x + 5y – 7
2x – 11 + y = 4x + 5y – 7
2x – 11 – 4y = 4x – 7
– 4y = 2x + 4
1
y   x  1 is in slope-intercept form.
2
The slope is 
Identify the slope:
1
.
2
4) Find the slope in the linear equation, 3y + 9 = 15.
Solve for y:
3y = 6
y = 2 is in slope-intercept form.
Identify the slope:
There is no "x" variable. However, it is understood to be there when we rewrite the equation as
y = 2 + 0x. Therefore, the slope is zero and we know the line is horizontal.
5) Find the slope in the linear equation, 5x - 2y = 10 - 2y.
Solve for y:
5x = 10
x=2
Identify the slope:
There is no "y" variable. However, it is understood to be there when we rewrite the equation as
1
2
1
0y + x = 2. When we solve for y, the equation is y   x  and the slope is  which is
0
0
0
undefined. Therefore, when the equation does not have a "y" variable to solve for, the slope is
undefined and the line will be vertical.
SLOPES OF PARALLEL AND PERPENDICULAR LINES
DEFINITIONS:
Two lines are parallel if the two lines never intersect.
Two lines are perpendicular if the two lines form a right angle (90 degrees).
THEOREM:
1) If two lines are parallel, then they have equal slopes (m1 = m2).
Example: L1: 2x - 3y = 14 (y = 2/3x - 14/3) then m = 2/3
L2: y = 2/3x + 5 then m = 2/3
Lines L1 and L2 are parallel.
2) If two lines are perpendicular, then the slopes are negative reciprocals,
(m1∙ m2 = -1).
Example: L1: 2x - 3y = 1 (y = 2/3x - 1/3) then m = 2/3
L2: y = -3/2x + 6 then m = -3/2
to check if slopes are negative reciprocals, 2/3 ∙ -3/2 = -1.
Lines L1 and L2 are perpendicular.