Transcript Chapter 7 Notes - Dripping Springs Independent School District

Chapter 7 Notes
Writing Algebraic
Expressions
Refer to operation word
bank handout from earlier.
Read statement, identify key
words that identify the
operation.
Translate numbers and
operations into algebraic
expression.
Translate into an algebraic
expression
Five more than triple a
number.
Six less than half a number.
Four times the sum of a
number and eight.
Translate into an algebraic
expression
Five more than triple a
number. 5 + 3n
Six less than half a number.
½n–6
Four times the sum of a
number and eight. 4(n + 8)
Translate into an algebraic
expression
The price for six roses plus a
delivery fee of $15 if roses are
sold individually.
The total cost for text
messages if you are charged
$0.75 per message.
Translate into an algebraic
expression
The price for six roses plus a
delivery fee of $15 if roses are
sold individually. 6r + 15
The total cost for text
messages if you are charged
$0.75 per message. 0.75m
Translate into an algebraic
expression
You have $30 to spend. You
buy “b” books at $7 each.
What is the cost for the books?
How much change will you
receive?
Translate into an algebraic
expression
You have $30 to spend. You
buy “b” books at $7 each.
What is the cost for the books?
7b
How much change will you
receive? 30 – 7b
Translate into an algebraic
expression
Mrs. Mathews is “x” years old.
Mrs. Gibbs is eight more than
half as old as Mrs. Mathews.
Coach Van is 3 years younger
than Mrs. Gibbs.
Write an expression for Mrs. Gibbs
age.
Write an expression for Coach Van’s
age.
Write an expression that represents
the sum of all their ages.
Translate into an algebraic expression
Mrs. Mathews is “x” years old.
Mrs. Gibbs is eight more than half
as old as Mrs. Mathews. Coach
Van is 3 years younger than Mrs.
Gibbs.
Write an expression for Mrs. Gibbs
age. ½ x + 8
Write an expression for Coach Van’s
age. ½ x + 8 – 3 = ½ x + 5
Write an expression that represents
the sum of all their ages.
x+½x+8+½x+5
 2x + 13
Translate into an algebraic
expression
The cafeteria sells pizza slices
for $2 and chocolate chip
cookies for $0.50.
Write an expression to buy “p”
slices of pizza and “c” cookies.
How much change would you
get from $20?
Translate into an algebraic
expression
The cafeteria sells pizza slices
for $2 and chocolate chip
cookies for $0.50.
Write an expression to buy “p”
slices of pizza and “c” cookies.
2p + 0.50c
How much change would you
get from $20? 20 – (2p + 0.50c)
7-1 Solving 2-Step
Equations
To solve a 2-step equation,
you will use the properties of
equality. (whatever you do to one
side you must do to the other.)
Steps:
 first undo addition or
subtraction.
Then undo multiplication or
division.
Examples
3n - 6 = 15
+6 +6
3n = 21
3
3
N=7
Examples
15x + 3 = 48
R/4 - 10 = (-6)
Examples - answers
15x + 3 = 48

-3 -3
15x = 45
 15 15
 x=3
r/4 - 10 = (-6)

+10 + 10
r/4 = 4
4( r/4 ) = 4 (4)
 r = 16
Examples
b/3 + 13 = 11
9g + 11 = 2
Examples - answers
b/3 + 13 = 11

-13
-13
b/3 = -2
3(b/3) = 3(-2)
b = -6
9g + 11 = 2

- 11 - 11
9g = -9
9
9
g = -1
Negative Coefficients
Examples:
5 - x = 17
-a + 6 = 8
Negative Coefficients
Examples:
5 - x = 17
-5
-5
-x = 12
-1 -1
X = -12
-a + 6 = 8
 - 6 -6
-a = 2
-1 -1
a = -2
More Negative coefficients
-9 - y/7 = (-12)
13 - 6f = 31
More Negative coefficients
 -9 - y/7 = (-12)
 +9
+9
-y/7 = -3
 y/(-7) = -3
 (-7) (y/(-7)) = (-3)(-7)
 Y= 21
 13 - 6f = 31
 -13
-13
 -6f = 18
 -6
-6
 f = -3
Word Problems
Lynne wants to save $900 to
go to Puerto Rico. She saves
$45 each week and now has
$180. To find how many
more weeks w it will take to
have $900, solve 180 + 45w =
900.
Word Problems -Answers
 Lynne wants to save $900 to go to Puerto
Rico. She saves $45 each week and now
has $180. To find how many more weeks
w it will take to have $900, solve 180 +
45w = 900.
 180 + 45w = 900
 -180
- 180
 45w = 720
 45
45
 w=16
It will take 16 weeks.
Writing Equations
Read the problem.
 Determine what you need to
find.
Define the variable you intend to
use for the unknown.
Write the equation then solve.
Answer the problem using words
and units.
Check your solution for accuracy
and reasonableness.
Word Problem 1
Mike has $23. This is seven
more than twice what Julie
has. How much money does
Julie have?
Word Problem 1
Mike has $23. This is seven
more than twice what Julie
has. How much money does
Julie have
Let J = Julie
2j + 7 = 23
 -7 -7
2j = 16
2 2
J = 8
Julie has $8
Word Problem 2
Brianna ran 45 miles over
three day (Tues-Thurs). This is
five less than four times the
miles she ran Monday. How
many miles did Brianna run
on Monday?
Word Problem 2
Brianna ran 45 miles over three
day (Tues-Thurs). This is five less
than four times the miles she ran
Monday. How many miles did
Brianna run on Monday?
Let m = miles
 4m – 5 = 45

+ 5 +5
 4m = 50
 4
4
 M = 12.5
Brianna ran 12.5 miles
Word Problem 3
The temperature in Chicago was
-7 degrees on Friday. Two weeks
ago the temperature was 8
degrees more than half Friday’s
temperature. That was the
temperature two weeks ago?
Word Problem 3
 The temperature in Chicago was -7
degrees on Friday. Two weeks ago the
temperature was 8 degrees more than
half Friday’s temperature. That was the
temperature two weeks ago?
 Let T=temperature
½ T + 8 = -7

-8 -8
 ½ T = - 15
 2( ½ T) = 2(-15)
 T = -30
The temp. was -30 degrees.
7-2 Solving Multi-Step
Equations
Combine like terms to
simplify an equation before
you solve it.
Then solve -- undo addition
or subtraction. Then multiply
or divide.
Combining Like Terms
M + 2M - 4 = 14

3M - 4 = 14
+4 +4
3M
= 18
3
3
M = 6
Example:
7 – y + 5y = 9
Finding Consecutive
Integers
Consecutive integers = when you
count by 1’s from any integer (ex.
120, 121, 122, 123)
Example: The sum of 3
consecutive integers is 96
N + (N+1) + (N+2) = 96
Using the Distributive
Property
2(5x - 3) = 14
38 = (-3)(4y + 2) + y
-3(m - 6) = 4
3(x + 12) - x = 8
Using the Distributive
Property - Answers
2(5x - 3) = 14
X=2
38 = (-3)(4y + 2) + y
y= -4
-3(m - 6) = 4
m= 4 2/3
3(x + 12) - x = 8
x= -14
7-3 Multi-Step Equations
with Fractions and
Decimals
When there is a fraction next
to a variable, you can do the
reciprocal to solve the
equation
Examples
2 n - 6 = 22
3
-(7/10)k + 14 = (-21)
2/3(m - 6) = 3
Examples - Answers
2 n -6 = 22
3
n= 42
-(7/10)k + 14 = (-21)
k=50
2/3(m - 6) = 3
m=10 1/2
Word Problems
Suppose your cell phone plan
has $20 per month plus $0.15
per minute. Your bill is
$37.25. Use the equation
20 + 0.15x = 37.25. How
many minutes are on your
bill?
Word Problems Answers
Suppose your cell phone plan
has $20 per month plus $0.15
per minute. Your bill is
$37.25. Use the equation
20 + 0.15x = 37.25. How
many minutes are on your
bill?
x=115
Consecutive Integers
When you count by 1’s from any
integer, you are counting
consecutive integers
Example: 45, 46, 47
When you count by 2’s from any
number you are counting either
consecutive odd or even
integers
Example: 2, 4, 6 or 3, 5, 7
Finding Consecutive
Integers
The sum of 3 consecutive integers is
96. Find the numbers.
Find two consecutive even integers
with a sum of 66.
Find 2 consecutive even integers
such that the sum of the larger
and twice the smaller is 38.
Finding Consecutive
Integers - Answers
The sum of 3 consecutive integers is
96. Find the numbers.
n + (n+1) + (n+2)=96; 31, 32, 33
Find two consecutive even integers
with a sum of 66.
n + (n+2) = 66; 32,34
Find 2 consecutive even integers
such that the sum of the larger
and twice the smaller is 38.
2n + (n +2) = 38; 12, 14
7-4 Write an Equation
Five times a number decreases by
11 is 9.
Find the number such that three
times the number increased by 7 is
52.
Find a number such that seven less
than twice the number is 43.
7-4 Write an Equation Answers
Five times a number decreases by
11 is 9.
5n - 11 = 19 n = 6
Find the number such that three
times the number increased by 7 is
52.
3n + 7 = 52 n = 15
Find a number such that seven less
than twice the number is 43.
2n - 7 = 43 n = 25
Fifteen more than the
product of 8 and a number
is -17.
Negative three times a
number less four is 17.
The product of 5 and a
number increased by 10 is
145.
Answers
Fifteen more than the product
of 8 and a number is -17.
15 + 8n = -17 n = -4
Negative three times a
number less four is 17.
-3n - 4 = 17 n = -7
The product of 5 and a
number increased by 10 is
145.
5n + 10 = 145 n= 27
The difference between
half a number and 9 is -23.
The quotient of a number and 5,
diminished by 11 is 18.
Answers
The difference between
half a number and 9 is -23.
1/2n - 9 = -23 n = -28
The quotient of a number and 5,
diminished by 11 is 18.
N/5 - 11 = 18 n = 145
Rachel hung 38 ornaments on
the tree. This is 3 less than half
what Jane hung on the tree.
How many ornaments did Jane
hang on the tree?
Sue did three more than twice
the amount of sit-ups that Lisa
did. If Sue did 67 sit-ups, how
many did Lisa do?
Answers
Rachel hung 38 ornaments on
the tree. This is 3 less than half
what Jane hung on the tree.
How many ornaments did Jane
hang on the tree?
1/2n - 3 = 38 n = 82
Sue did three more than twice
the amount of sit-ups that Lisa
did. If Sue did 67 sit-ups, how
many did Lisa do?
3 + 2n = 67 n = 32
Four friends go to dinner
together. The check totals
$36. They have a coupon for
$4 off the total bill. They
decide to split the check
equally. How much does
each person pay?
Answers
Four friends go to dinner
together. The check totals
$36. They have a coupon for
$4 off the total bill. They
decide to split the check
equally. How much does
each person pay?
4x +4 = 36
x = $8 each
Mrs. Mathews has 3 more
than twice the number of
Christmas pins that Ms.
Holden has. If Mrs. Mathews
has 39 pins, how many does
Ms. Holden have?
The price of regular set of
golf clubs was $179.95. The
sale price was $113.25. How
much do you save?
Answers
Mrs. Mathews has 3 more than
twice the number of Christmas
pins that Ms. Holden has. If Mrs.
Mathews has 39 pins, how many
does Ms. Holden have?
2x +3 = 39 x = 18 pins
The price of regular set of golf
clubs was $179.95. The sale price
was $113.25. How much do you
save?
113.25 + x = 179.95
x = $66.70
Word Problems
Two-thirds the number of girls
plus two represents the
number of boys in the class.
If there are 13 boys in the
class, how many girls are
there?
Word Problems Answers
Two-thirds the number of girls
plus two represents the
number of boys in the class.
If there are 13 boys in the
class, how many girls are
there?
2/3y + 2 = 13
y = 16.5
7-5 Solving Equations
with Variables on Both
Sides
To solve an equation with a
variable on both sides, use
addition or subtraction to
collect the variable on one
side of the equation.
Collecting the variable
on one side
9a + 2 = 4a - 18
4x + 4 = 2x + 36
k + 9 = 6(k - 11)
Collecting the variable
on one side - Answers
9a + 2 = 4a - 18
a = -4
4x + 4 = 2x + 36
x = 16
k + 9 = 6(k - 11)
k = 15
Word Problem
Beth leaves home on her bicycle,
riding at a steady rate of 8 mi/h.
Her brother, Ted, leaves home on
his bicycle 1/2 an hour later,
following Beth’s route. He rides
at a steady rate of 12 mi/h. How
long after Beth leaves home will
Ted catch up?
Word Problem - Answer
Beth leaves home on her bicycle,
riding at a steady rate of 8 mi/h.
Her brother, Ted, leaves home on
his bicycle 1/2 an hour later,
following Beth’s route. He rides
at a steady rate of 12 mi/h. How
long after Beth leaves home will
Ted catch up?
8x = 12(x - 1/2) x = 1.5
7-5 Solving Equations
with Variables on both
sides (Day 2)
5(w + 3) = 4(w - 2)
9 - d = -24 - 4d
7-5 Solving Equations
with Variables on both
sides (Day 2) - Answers
5(w + 3) = 4(w - 2)
w= -23
9 - d = -24 - 4d
d= -11
Word Problems
• Five more than three times a
number is the same as four less
than twice a number. Find the
number.
• Sixty-seven, decreased by four
times a number, is the same as
eight times a number, increased
by seven. Find the number.
Word Problems Answers
• Five more than three times a
number is the same as four less
than twice a number. Find the
number.
5 + 3y = 2y – 4; y= -9
• Sixty-seven, decreased by four
times a number, is the same as
eight times a number, increased
by seven. Find the number.
67 – 4a = 8a + 7; a = 5
Find the value of x and
the perimeter
The square and the triangle have
equal perimeters.
A. Find the value of x
B. Find the perimeter
(Square: Side is x-3)
(Triangle: Sides are x, x, and 8)
Find the value of x and
the perimeter
The square and the triangle have
equal perimeters.
A. Find the value of x
4(x - 3) = x + x + 8;
B. Find the perimeter
x = 10
p = 28
(Square: Side is x-3)
(Triangle: Sides are x, x, and 8)
Find the missing value
The Yellow Bus Company charges
$160 plus $80 per hour to rent a
bus. The Orange Bus Company
charges $200 plus $60 per hour.
A. For what number of hours would
the companies charge the
same?
B. What would the charge be for
that number of hours?
Find the missing value
The Yellow Bus Company charges
$160 plus $80 per hour to rent a
bus. The Orange Bus Company
charges $200 plus $60 per hour.
A. For what number of hours would
the companies charge the same?
160 + 80h = 200 + 60h; h = 2 hours
B. What would the charge be for that
number of hours? $320
7-7 Transforming
Formulas
You can use the properties of
equality to transform a
formula to represent one
quantity in terms of another.
Transforming in one step
Solve the area formula A = lw
for l
Examples: p = s - c (solve for s)
h = k/j (solve for k)
Transforming in one step
- Answers
Solve the area formula A = lw
for l
l = A/w
Examples: p = s - c s = p + c
h = k/j
k=hxj
Using more than one
step
Solve the formula P = 2L + 2W for L
Y = 3/5p - 4 solve for p
R = n(C - F) solve for C
Using more than one
step - Answers
Solve the formula P = 2L + 2W for L
l = (P-2w)/2
Y = 3/5p - 4 solve for p
p = 5/3(y + 4)
R = n(C - F) solve for C
C = (R + nF)/n