Slope

Slope: The steepness of a line expressed as the ratio of the vertical change in y to
the horizontal change in x.
To find the slope of a line through 2 given points, we say
rise y y 2  y1
m


run x x2  x1
3
Ex: Given point (-3, -2), graph the line having slope . Remember: up 3, right 2.
2
Ex: Given point (1,7), graph the line having slope
2
. Remember: down 2, right 3.
3
Ex: Given point (-4, 1), graph the line having a slope of 4.

When we are not given the slope, we can find the slope of the line given 2 points
on that line.
Ex: Find the slope of the line through points (6,-2) and (5,4).
Use the FORMULA!
m
y y 2  y1 4  (2) 6



 6
x x2  x1
56
1
Exs: Find the slope of the line passing through the points.
1) (-3,5) and (-4,-7)
2) (6,-8) and (-2,4)
3) (-8,4) and (2,4)
*****ALL HORIZONTAL LINES HAVE SLOPE 0!!!! m=0 *****
4) (6,2) and (6,-4)
***ALL VERTICAL LINES HAVE UNDEFINED SLOPE***

We can also find the slope of a line from the line’s equation.
Ex: y = -3x + 5
If we choose 2 values for x and find their corresponding
y’s, we can then find the slope of the line through 2 points
on the line.
Let’s let x = -2 and x = 4
y = -3 (-2) + 5
y = -3 (4) + 5
=6+5
= 11
(-2, 11)
= -12 + 5
= -7
(4, -7)
m
11  (7) 18

 3
24
6
Notice the coefficient of x in the equation of our line. The coefficient -3 matches our
slope for the same line. Therefore, we can conclude that the slope of the line is the
coefficient of x in the equation of the line.
Find the slope of the line on the following:
7
x 1
1) y 
2) 3x  2 y  9
3) y  4  0
2

4) x  3  7
We can also use slope to determine if 2 lines are parallel, perpendicular, or
neither.
x  2y  4
1
Notice that m =
x  2 y  6
2
for both equations. Therefore, NON-VERTICAL PARALLEL LINES ALWAYS
HAVE EQUAL SLOPES!!!!!!!!
Ex: Find the slope of each of the following lines.
x  2y  4
1
Notice m =
for
2x  y  6
2
equation 1 and m = 2 for equation 2. When the slopes of the two lines are the negative
reciprocals of one another (or their product = -1), the lines are PERPENDICULAR.
Ex: Find the slope of each of the following lines.
Ex: Decide if the following are parallel, perpendicular, or neither.
1)
3x  y  4
6 x  2 y  12
2)
4x  3y  6
2x  y  5
4)
3x  y  4
x  3y  9
5)
3 x  7 y  35
7 x  3 y  6
3)
5x  y  1
x  5 y  10