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Created by Charlean Mullikin: mullikinc@anderson3.k12.sc.us ML sections 3.6/3.7
Slope is the relationship of the rise to the run of a line.
m = rise = y2 – y1
run x2 – x1
Slope can be positive: + ÷ + or - ÷ -
Slope can be negative:
+ ÷ - or - ÷ +
Slope can be 0: 0
÷
anything
Slope can be undefined:
Anything
÷
0
Horizontal
Vertical
ALWAYS SIMPLIFY SLOPES
Slopes are positive, negative, 0, or Undefined (No slope).
Slopes are written as integers with one sign, proper fractions, or improper fractions (no mixed fractions).
When 0 is on top, the slope is 0.
When 0 is on bottom, the slope is undefined or no slope.
m = 5
m = -4
m = 1/3
m = - 3/5
m = 5/2
m = 0
m = undefined
m = 5 1/2
m = -5/-3
m = 15/3
m = -15/-25m = 0/6
m = 5/0
m = 12/-8
m = rise = y2 – y1
run x2 – x1
(x1 , y1)
(x2 , y2)y2x2
x1 y1
– –
RiseOn
top!!
Run On bottom!!
m = rise = y2 – y1 =
run x2 – x1
(3 , -3)
(0 , 9)90
3 -3–
–
RiseOn
top!!
Run On bottom!!
Find the slope of the line that passes through (3, -3)and (0 , 9)
-3
0= -12
3
– 9
3 –
= -4
a: m=+5
+5
+5
+5
=1
b: m=+2
+2
+2
+2 =1
YES, Since the slopes are the same (1=1),
then the lines ARE PARALLEL.
6/2 = 3 -10/-2 = 5 -24/8 = -3
2/6 = 1/3 9/0 = undefined 0/22 = 0
ApplicationApplication
Identify rise and run.Which word points to the rise?
Put the rise on top.
What is the run?
Put the run on bottom.
Change to same units, then Divide out and Answer the question in reasonable units.
3600 feet
3.1 miles=
The average slope is about .22.
3.1 x 5280 = 16368 ft
16328 feet
Perpendicular LinesPerpendicular Lines When two lines are perpendicular, there When two lines are perpendicular, there
are two cases with relation to slopes:are two cases with relation to slopes: Case 1-If neither line is vertical, the Case 1-If neither line is vertical, the
product of the two slopes is negative one product of the two slopes is negative one (Opposite reciprocals). (Opposite reciprocals). mm11=2/3 and m=2/3 and m22= - 3/2= - 3/2
Case 2 – If one of the lines is vertical, Case 2 – If one of the lines is vertical, then the perpendicular line is horizontal. then the perpendicular line is horizontal. mm11=undefined and m=undefined and m22= 0= 0
What is the slope of…..What is the slope of…..
Slope of given line Parallel Line? Perpendicular Line?
1/2
-6
3/5
-8/7
0
4
No slope
1/2
-6
3/5
-8/7
0
4
No slope
-2
1/6
-5/3
7/8
No slope
-1/4
0
Writing EquationsWriting Equations
Shortcut #1
1
Writing EquationsWriting Equations
Shortcut #2
1
Writing EquationsWriting Equations
Writing EquationsWriting Equations
Shortcut #1
Shortcut #2
Writing EquationsWriting Equations
1 1
Identify ONE point to useFind slope
Substitute
Simplify and solve for y
Distributive Property of =
Addition Property of = (Add 8 to both sides)
Combine like termsUse calculator!
Parallel EquationsParallel Equations
Lines that are parallel have the same Lines that are parallel have the same slope.slope.– Identify slope of given lineIdentify slope of given line– Identify point parallel line passes Identify point parallel line passes
throughthrough– Use point-slope equation to write Use point-slope equation to write
equationequation
Parallel EquationsParallel Equations
Write the equation of the line parallel Write the equation of the line parallel to y = ¾ x – 5 that passes through the to y = ¾ x – 5 that passes through the point (3, -2).point (3, -2).m = ¾, parallel slope is also ¾m = ¾, parallel slope is also ¾Point (3, -2)Point (3, -2)y – yy – y11 = m(x – x = m(x – x11))y - -2 = ¾(x – 3)y - -2 = ¾(x – 3)y + 2 = ¾ x – 9/4y + 2 = ¾ x – 9/4y = ¾ x – 9/4 – 2 y = ¾ x – 9/4 – 2 y = ¾ x – 17/4y = ¾ x – 17/4
Parallel EquationsParallel Equations Write the equation of the line parallel to Write the equation of the line parallel to 7x + 5y = 13 that passes through the point 7x + 5y = 13 that passes through the point (1, 2).(1, 2).
Solve for y to find slope:Solve for y to find slope: 7x + 5y = 137x + 5y = 13 5y = -7x + 13 (subtract 7x from both sides)5y = -7x + 13 (subtract 7x from both sides) y = -7/5 x + 13/5 (Divide each term by 5)y = -7/5 x + 13/5 (Divide each term by 5) parallel slope is – 7/5parallel slope is – 7/5
Point (1, 2)Point (1, 2) y – yy – y11 = m(x – x = m(x – x11)) y - 2 = - 7/5 (x – 1)y - 2 = - 7/5 (x – 1) y - 2 = -7/5 x + 7/5y - 2 = -7/5 x + 7/5 y = -7/5 x + 7/5 + 2 y = -7/5 x + 7/5 + 2 y = -7/5 x + 17/5y = -7/5 x + 17/5
Perpendicular EquationsPerpendicular Equations
Lines that are perpendicular have Lines that are perpendicular have slopes that multiply to equal -1. They slopes that multiply to equal -1. They are opposite sign, reciprocal are opposite sign, reciprocal numbers.numbers.– Identify slope of given lineIdentify slope of given line– Change the sign and flip the number to Change the sign and flip the number to
get the perpendicular slope.get the perpendicular slope.– Use point-slope equation to write Use point-slope equation to write
equationequation
Perpendicular EquationsPerpendicular Equations Write the equation of the line perpendicular Write the equation of the line perpendicular
to to 7x + 5y = 13 that passes through the point 7x + 5y = 13 that passes through the point (1, 2).(1, 2).
Solve for y to find slope:Solve for y to find slope: 7x + 5y = 137x + 5y = 13 5y = -7x + 135y = -7x + 13 y = -7/5 x + 13/5y = -7/5 x + 13/5 perpendicular slope is +5/7perpendicular slope is +5/7
Point (1, 2)Point (1, 2) y – yy – y11 = m (x – x = m (x – x11)) y - 2 = +5/7(x – 1)y - 2 = +5/7(x – 1) y - 2 = 5/7 x – 5/7y - 2 = 5/7 x – 5/7 y = 5/7 x – 5/7 + 2 y = 5/7 x – 5/7 + 2 y = 5/7 x + 9/7y = 5/7 x + 9/7
Perpendicular EquationsPerpendicular Equations
Write the equation of the line Write the equation of the line perpendicular to y = ¾ x – 5 that perpendicular to y = ¾ x – 5 that passes through the point (3, -2).passes through the point (3, -2).m = ¾, perpendicular slope is – 4/3 m = ¾, perpendicular slope is – 4/3 Point (3, -2)Point (3, -2)y – yy – y11 = m(x – x = m(x – x11))y - -2 = -4/3(x – 3)y - -2 = -4/3(x – 3)y + 2 = -4/3 x + 4y + 2 = -4/3 x + 4y = -4/3 x + 4 – 2 y = -4/3 x + 4 – 2 y = -4/3 x + 2y = -4/3 x + 2
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