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Chapter 7 Notes
7-1 Solving 2-Step
Equations
To solve a 2-step equation,
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)
B/3 + 13 = 11
9g + 11 = 2
Examples - Answers
15x + 3 = 48
x=3
R/4 - 10 = (-6)
R=16
B/3 + 13 = 11
B= -6
9g + 11 = 2
g= -1
Negative Coefficients
Examples:
5 - x = 17
-a + 6 = 8
-9 - y/7 = (-12)
13 - 6f = 31
Negative Coefficients Answers
 Examples:
 5 - x = 17
x=(-12)
 -a + 6 = 8
a=(-2)
 -9 - y/7 = (-12)
y=(21)
 13 - 6f = 31
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.
w=16
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
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
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
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
7-8 Simple and
Compound Interest
When you first deposit money in a
savings account, your deposit is
called principal.
The bank takes the money and
invests it. In return, the bank pays
you interest based on interest
rates.
Simple interest is interest paid only
on the principal.
Simple interest formula
I = prt
I = interest
P = principal
R = interest rate per year
T = time in years
Simple Interest I = prt
I = interest
Interest
Principal
Rate
Time
$ 200
4%2 years
$ 850
6%5 years
$ 1,200
4%30 months
p = principal t = time in years r = rate
Work Amount of Interest Balance
Simple Interest
I = prt
I = interest
Interest
Principal
Rate
Time
p = principal t = time in years r = rate
Work
Amount of Interest
Balance
$ 200
4% 2 years
200(.04)(2)
16 200+16=216
$ 850
6% 5 years
850(.06)(5)
255850+255=1105
$ 1,200
4% 30 months
1200(.04)(30/12)
1201200+120=1320
Compound interest
When a bank pays interest on the
principal and on the interest an
account has earned, the bank is
paying compound interest.
The principal plus the interest is the
balance, which becomes the
principal on which the bank
figures the next interest payment.
Complete the table. Compound the interest annually
$3000 at 5% for 4 years
Balance at beginning
Interest
of year
Yr1:
Yr2:
Yr3:
Yr4:
Balance
Complete the table. Compound the interest annually
$3000 at 5% for 4 years
Balance at beginning
Interest
of year
Balance
Yr1:
30003000(.05)(1)=150
3000+150=3150
Yr2:
31503150(.05)(1)=157.50
3150+157.50=3307.50
Yr3:
3307.5 3307.50(.05)(1)=165.38
3307.50+165.38=3472.88
Yr4:
3472.883472.88(.05)(1)=173.64
3472.88+173.64=3646.52
Compound Interest
Formula
Formula --- B = p(1 + r)n
B = final balance
P = principal
R = interest rate for each
interest period
N = number of interest periods
Interest and Interest
Periods
Semiannually = 2 times a year
Semiannually for 2 years = 4
interest period
Semiannually at 3% = 3/2 = 1.5
n
Compound Interest B=p(1 + r)
B= balance
p = principal
r= interest rate
n = number of interest periods
Find the balance
Interest
Principal Rate
Compounded
Time (years)
Work
$495
4% annually
2 years
$1,280
3% anually
3 years
$15,600
3% semiannually
3 years
$2,000
2% semiannually
2 years
Balance
n
Compound Interest B=p(1 + r)
B= balance
r= interest rate
Find the balance
Interest
Principal Rate
Compounded
$495
$1,280
$15,600
$2,000
4% annually
3% anually
3% semiannually
2% semiannually
p = principal
n = number of interest periods
Time (years)
2 years
3 years
3 years
2 years
Work
2
495(1.04)
Balance
535.39
3
1280(1.03)
1398.69
6
15600(1.015)
4
2000(1.01)
17057.71
2081.21