Download 5.5 Parallel and Perpendicular

Parallel and
Perpendicular Lines
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The Dark Doodad Nebula
Warm-Up
1.Find the reciprocal of each fraction
a.
1
2
b.
4
3
c.
1

3
d.
5

2
2.Find the slope and y-intercept of each equation
5
a. y  x  4
4
5
b. y  x  8
3
c. y  6 x
d. y  6 x  2
Quick Review/Preview
1. What is the Slope-Intercept Form?
y = mx + b
2. What is the Standard Form?
Ax + By = C
3. What is the Point-Slope Form?
y – y1 = m(x – x1)
4. Find the “negative reciprocal” of each fraction
2
5
a.

5
2
3 7
b. 
7 3
1 2
4
c. 
 2 d. 
2 1
3
3
4
Overview
Parallel Lines and Perpendicular Lines
 Property of their slopes
 Write equations for each type of lines

Parallel Lines
Parallel Lines - lines that never intersect
What is the slope of the
red line?
1/2
What is the slope of the
blue line?
1/2
Parallel Lines have the same slope!
Parallel Lines
Determine if the lines are parallel
1
1. 2 x  6 y  12 and y   x  5
3
2x + 6y = 12
6y = -2x + 12
1
y  x2
3
yes
Parallel Lines
Determine if the lines are parallel
3
2. 6 x  8 y  24 and y 
x 3
4
6x + 8y = -24
8y = -6x – 24
3
y   x 3
4
No
Parallel Lines
Determine if the lines are parallel
2
3. 4 x  6 y  2 and y  x  8
3
4x + 6y = -2
6y = -4x – 2
2
1
y  x
3
3
No
Equations of Parallel Lines
4. Write an equation for the line that
3
contains (5, 1) and is parallel to y  x  4
5
3
m=
5
3
y  1   x  5
5
Equations of Parallel Lines
5. Write an equation for the line that
contains (2, -6) and is parallel to 3x  y  9
m=3
y + 6 = 3(x – 2)
Equations of Parallel Lines
6. Write an equation for the line that
1
contains (-4, 3) and is parallel to y  x  7
2
1
m=
2
1
y  3  x  4
2
Perpendicular Lines
Perpendicular Lines – lines that intersect to form
right angles
What is the slope of the
red line?
-1/4
What is the slope of the
blue line?
4/1
Perpendicular Lines have negative reciprocal slope!
Perpendicular Lines
What is the slope of the perpendicular line?
2
7. y  x  8
5
1
8. y   x
5
9. 2 x  y  7
-5/2
5/1 = 5
1/2
Equations of Perpendicular Lines
10. Find the equation of the line that
contains (0, -2) and is perpendicular to
y = 5x + 3
1
m=
5
1
y2 x
5
Equations of Perpendicular Lines
11. Find the equation of the line that
contains (1, 8) and is perpendicular to
3
y  x 1
4
4
m =
3
4
y  8    x  1
3
Equations of Perpendicular Lines
12. Find the equation of the line that
contains (2, -3) and is perpendicular to
x  2y  6
m=2
y + 3 = 2(x – 2)
Perpendicular Lines
13. Determine if the lines are perpendicular:
2
y  x  1 and 3 y  2 x  4
3
3y  2x  4
3 y  2 x  4
2
4
y  x
3
3
No
14. Determine if the lines are perpendicular:
4
y  x  5 and 4 y  3 x  9
3
4y  3x  9
4 y  3 x  9
3
9
y  x
4
4
Yes
Challenge
What is the slope of the line that is parallel
to x = 4?
undefined
 What is the slope of the line that is
perpendicular to x = 4?
zero

Graphing Lines

Y = 3x – 2

2x – 5y = 15
Lesson 2 Warm Up

1. Find the equation of the line that is // to
3x + 4y = 8 through the point (8, -2).

2. Find the equation of the line that is
perpendicular to 3x – y = 4 through the
point (-9, 3).
Lesson 2: Systems
What is a system of equations?
 What are the 3 methods we can use to
solve a system?

Possible solutions when you
have 2 equations
Solution by Graphing
𝑥 + 2𝑦 = −7
2𝑥 − 3𝑦 = 0
Elimination
1.
4𝑥 − 2𝑦 = 7
𝑥 + 2𝑦 = 3
2.
4𝑥 + 3𝑦 = 4
2𝑥 − 𝑦 = 7
Substitution

𝑦 = −𝑥 − 12
𝑦 = 2/3𝑥 − 2
Special Cases
2𝑥 − 3𝑦 = 18
−2𝑥 + 3𝑦 = −6
2𝑥 − 𝑦 = 3
−2𝑥 + 𝑦 = −3
Lesson 3 – Systems with 3
variables
Try plotting these points
in the 3-D plane.
(4, 2, 1)
(3, -2, 4)
(5, 0, -1)
Solve with Elimination

3𝑥 + 2𝑦 + 4𝑧 = 11
2𝑥 − 𝑦 + 3𝑧 = 4
5𝑥 − 3𝑦 + 5𝑧 = −1

−𝑥 − 5𝑦 + 𝑧 = 17
−5𝑥 − 5𝑦 + 5𝑧 = 5
2𝑥 + 5𝑦 − 3𝑧 = −10

−6𝑥 − 2𝑦 + 2𝑧 = −8
3𝑥 − 2𝑦 − 4𝑧 = 8
6𝑥 − 2𝑦 − 6𝑧 = −18

𝑥 − 2𝑦 + 3𝑧 = 9
𝑦 + 3𝑧 = 5
2𝑧 = 4
𝑥+𝑦+𝑧 =6
2𝑥 − 𝑦 + 𝑧 = 3
3𝑥 − 𝑧 = 0

Suppose you have saved $3,200 from a parttime job, and you want to invest your savings in
a growth fund, an income fund, and a money
market fund. To maximize your return, you
decide to put twice as much money in the growth
fund as in the money market fund. Your return
on investment will be 10% of the growth fund,
7% of the income fund, and 5% of the money
market fund. How should you invest the $3,200
to get a return of $250 in one year?

A theater has tickets at $6 for adults,
$3.50 for students, and $2.50 for children
under 12 years old. A total of 278 tickets
were sold for one showing with a total
revenue of $1,300. If the number of adult
tickets sold was 10 less than twice the
number of student tickets, how many of
each type of ticket were sold for the
showing?
Day 4 – Solving with Matrices

𝑥 − 3𝑦 + 3𝑧 = −4
2𝑥 + 3𝑦 − 𝑧 = 15
4𝑥 − 3𝑦 − 𝑧 = 19
𝑥 + 𝑧 − 2𝑦 = −4
𝑦 − 2𝑧 = 1 + 4𝑥
−𝑧 = 10 − 2𝑥 − 2𝑦

The perimeter of a triangle is 19 cm. If the
length of the longest side is twice that of the
shortest side and 3 cm less than the sum of the
lengths of the other two sides, find the lengths of
the three sides.

The measure of the largest angle of a
triangle is 10°more than the sum of the
measures of the other two angles and 10°
less than 3 times the measure of the
smallest angle. Find the measures of the
three angles of the triangle.

Jacksonville, Florida has an elevation of
12 ft. above sea level. A hot air balloon
taking off from Jacksonville rises 50
ft./min. Write an equation to model the
balloon’s elevation as a function of time.
A candle is 6 in. tall after burning for 1
hour. After 3 hours, it is 5.5 in. tall. Write
a linear equation to model the height y of
the candle after burning for x hours.
 Step 1: Identify your data points. Step 2:
Find the slope of the line.
 Step 3. Use one of the points and the
point-slope form
__________________________ to write
an equation for the line.


A woman is considering buying a car built
in 1999. She researches prices for
various years of the same model and
records the data in a table.
Model
Year
Prices
2000
2001
2002
2003
2004
$5,784
$6,810
$8,237
$9,660
$10,948
Lesson 5 Linear Programming