The document discusses slopes and equations of lines. It defines slope as rise over run and explains how to calculate slope from two points on a line or from the graph of a line. It also explains how to write the equation of a line in point-slope form, slope-intercept form, or for horizontal and vertical lines. Several examples are provided of finding slopes and writing line equations in different forms.

1. Slopes and Equations of Lines
The student is able to (I can):
• Find the slope of a line given
– Graph
– Two points
– Equation
• Write the equation of a line given
– Slope and y-intercept
– Slope and point on line
– Two points
– Graph

2. riseriseriserise – the difference in the yyyy----valuesvaluesvaluesvalues of two points on a line.
runrunrunrun – the difference in the xxxx----valuesvaluesvaluesvalues of two points on a line.
•
•(xxxx1111, yyyy1111)
(xxxx2222, yyyy2222)
run =run =run =run = 3333
rise =rise =rise =rise = 5555

3. slopeslopeslopeslope – the ratio of riseriseriserise to runrunrunrun. If (xxxx1111, yyyy1111) and (xxxx2222, yyyy2222) are any
two points on a line, the slope of the line is
So, for the previous example, substitute (2222, 1111) for (xxxx1111, yyyy1111) and
(5555, 6666) for (xxxx2222, yyyy2222):
Always reduce fractions to their simplest forms. Also,
it’s usually best to leave improper fractions improper.
2 1
2 1x
y
m
y
x
−
−
=
2 1
2 1 5
5
6
2
1
3x x
y
m
y− −
= = =
− −

4. Summary: Slope of a Line
positive slope negative slope
zero slope undefined slope

5. Practice
Find the slopes between the following points:
1. (2, –1) and (8, –4) 2. (–3, 10) and (5, –6)
3. (1, 12) and (–10, –10) 4. (22, 4) and (0, 28)

6. Practice
Find the slopes between the following points:
1. (2, –1) and (8, –4) 2. (–3, 10) and (5, –6)
3. (1, 12) and (–10, –10) 4. (22, 4) and (0, 28)
( )4 1 3 1
8 2 6 2
m
− − − −
= = = −
− ( )
6 10 16
2
5 3 8
m
− − −
= = = −
− −
10 12 22
2
10 1 11
m
− − −
= = =
− − −
28 4 24 12
0 22 22 11
m
−
= = = −
− −

7. pointpointpointpoint----slope formslope formslope formslope form – given the slope, mmmm, and a point on the line
(xxxx1111, yyyy1111), the equation of the line is
y – yyyy1111 = mmmm(x – xxxx1111)
Example: Write the equation of the line whose slope is 2222,
which goes through the point (1111, 6666)
y – 6666 = 2222(x – 1111)
In point-slope form, you can leave it like this – you don’t have
to simplify it any further.

8. slopeslopeslopeslope----intercept formintercept formintercept formintercept form – given the slope, mmmm, and bbbb, the y-
intercept, the equation of the line is
y = mmmmx + bbbb
Example: For mmmm = ––––3333 and y-intercept 7777, find the equation of
the line.
y = ––––3333x + 7777

9. horizontalhorizontalhorizontalhorizontal linelinelineline – for a horizontal line (mmmm = 0000), the equation of
the line is
y = bbbb
vertical linevertical linevertical linevertical line – for a vertical line (mmmm = undefinedundefinedundefinedundefined), the equation
of the line is
x = xxxx1111
Notice that this equation does not start with “y =“

10. Practice
Write the equations in point-slope form
1. m = –2; (3, 5) 2. (1, 2)
3. m = –4; (–1, 4) 4. (–2, –3)
Write the equations in slope-intercept form
5. m = 6, b = 2 6. m = –1, b = –4
2
;
3
m =
3
;
4
m = −

11. Practice
Write the equations in point-slope form
1. m = –2; (3, 5) 2. (1, 2)
y – 5 = –2(x – 3)
3. m = –4; (–1, 4) 4. (–2, –3)
y – 4= –4(x + 1)
Write the equations in slope-intercept form
5. m = 6, b = 2 6. m = –1, b = –4
y = 6x + 2 y = –x – 4
2
;
3
m =
3
;
4
m = −
2
2 ( 1)
3
y x− = −
3
3 ( 2)
4
y x+ = − +

12. Practice
7. Write the equation of the line through (––––1111, 6666) and (1, 2) in
slope-intercept form.

13. Practice
7. Write the equation of the line through (––––1111, 6666) and (1, 2) in
slope-intercept form.
2 6 4
1 ( 1) 2
2m
− −
= = =
−
−
−
Method #1
y – 6666 = ––––2222(x + 1111)
y = –2x – 2 + 6
y = –2x + 4
Method #2
6666 = (––––2222)(––––1111) + b
6 = 2 + b
4 = b
y = –2x + 4