Download Chapter 3 - El Camino College

Chapter 3. Language of Algebra
3.1
3.2
3.3
3.4
3.5
3.6
Algebraic Expressions
Evaluating Algebraic Expressions and Formulas
Simplifying Algebraic Expressions and the
Distribution Property
Combine Like Terms
Simplifying Expressions to Solve Equations
Using Equations to Solve Application Problems
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3.1 Algebraic Expressions
1. Translate word phrases to algebraic expressions
2. Write algebraic expressions to represent unknown
quantities
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3.1 Algebraic Expressions
1. Translate word phrases to algebraic expressions
Variable: a letter (or symbol) that stands for a number.
Algebraic Expressions
Variables and/or numbers can be combined with
operations of addition, subtraction, multiplication and
division to created algebraic expression.
e.g. 4a + 7,
10 − y
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3.1 Algebraic Expressions
1. Translate word phrases to algebraic expressions
Addition
Subtraction
the sum of a and 8
a+8
the difference of 23 and P
23 – P
4 plus c
16 added to m
4 more than t
20 greater than F
T increased by r
exceeds y by 35
4+c
m + 16
t+4
F + 20
T+r
y + 35
550 minus h
18 less than w
7 decreased by j
M reduced by x
12 subtracted from L
5 less f
550 – h
w – 18
7–j
M–x
L – 12
5–f
compare “18 less than w” and “5 less f”
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3.1 Algebraic Expressions
1. Translate word phrases to algebraic expressions
Multiplication
the product of 4 and x
20 times B
twice r
double the amount a
triple the profit P
three-fourth of m
Division
4x
20B
2r
2a
3P
3
4
m
the quotient of R and 19
R
19
s divided by d
s
d
k split into 4 parts
k
4
the ratio of c to d
c
d
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3.1 Algebraic Expressions
1. Translate word phrases to algebraic expressions
e.g.1 Write each phrase as an algebraic expression
Ans. p – 15
a.
15 less than the population p
b.
Twice the weight w
Ans. 2w
c.
The product of the volume V and 3, increased by 60
Ans. 3V + 60
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3.1 Algebraic Expressions
2. Write algebraic expressions to represent unknown quantities
e.g.2 PACKAGING A regular-size package of Oreos cookies
weights x ounces. A family-size package of Oreos weights 16
ounces more. Write an algebraic expression that represents
the weight (in ounces) of the family-size pack.
Ans. x + 16
e.g.3 JEWELRY The value of a wedding ring is three times that of
an engagement ring. Choose a variable to represent the value
of one of the rings. Then write an algebraic expression that
represent the value of the other ring.
Ans. x = value of the engagement ring
Then, 3x = value of the wedding ring
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3.1 Algebraic Expressions
2. Write algebraic expressions to represent unknown quantities
e.g.4 CAMPING A rope is to be cut into six equally long pieces.
Write an algebraic expression that represents the length of
each piece.
let r = length of the rope in feet.
Then, length of each piece (in ft) =
r
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e.g.5 VEHICLE WEIGHTS The weight of a Cadillac Escalade is
900 lb less than four times a Smart Car. Choose a variable to
represent the weight of one of he cars. Write an algebraic
expression that represents the weight of the other car.
Let x (in lb) = weight of a Smart Car,
then weight of a Cadillac (in lb) = 4x – 900
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3.1 Algebraic Expressions
2. Write algebraic expressions to represent unknown quantities
e.g.6 POLICE FORCE Part of a 500-officer police force works in
the field. The others have desk jobs. Choose a variable to
represent the number of officers that work in the field. Then
write an algebraic expression that represents the number of
that work force at a desk.
Let x = number of officers that work in the field,
then number of officers that work at a desk = 500 – x
e.g.7 LEADERS Hillary Clinton was born 7 years before Oprah
Winfrey. Michelle Obama was born 10 years after Oprah.
Write algebraic expressions to represent the ages of each of
these famous women.
Let x = the age of Oprah,
then the age of Hillary = x + 7, the age of Michelle = x – 10
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3.1 Algebraic Expressions
2. Write algebraic expressions to represent unknown quantities
e.g.8 Write an algebraic expression that represents the number of
hours in d days.
Ans. 24d hours
Comments: introducing tables, and inductive method
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3.2 Evaluating Algebraic Expressions and Formulas
1. Evaluating algebraic expressions
2. Use formulas from business to solve application
problems
3. Use formulas from science to solve application
problems
4. Find the mean (average) of a set of values
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3.2 Evaluating Algebraic Expressions and Formulas
1. Evaluating algebraic expressions
e.g.2 Evaluate each expression for x = 4:
Ans. 22
a.
6x – 2
− x −8
Ans. –4
b.
3
e.g.3 Evaluate each expression for x = –4:
Ans. 44
a.
2x2 – 3x
Ans. –17
b.
–x + 7(1 + x)
Ans. –89
c.
x3 – 25
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3.2 Evaluating Algebraic Expressions and Formulas
1. Evaluating algebraic expressions
e.g.4 Evaluate each expression for c = –3 and d = 9:
a.
( 2cd + 5d )2 Ans. 81
b.
| –8d – 3c | Ans. 63
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3.2 Evaluating Algebraic Expressions and Formulas
2. Use formulas from business to solve application
problems
e.g.5 FILMS It cost DreamWorks SKG about $528 million
to make and distribute the film Shrek. If the studio
has received approximately $916 million to date in
worldwide revenue from the film, find the profit the
studio has made on this movie.
Ans. $388 millions
Formula relating profit with revenue and cost.
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3.2 Evaluating Algebraic Expressions and Formulas
3. Use formulas from science to solve application
problems
e.g.6 SPEED LIMITS Texas’s daytime speed limit for cars
on rural interstate highways is 75 mph. How far
would a car travel in 4 hours at this speed?
Ans. 300 mi
Distance-Speed-Time formula: d = r t
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3.2 Evaluating Algebraic Expressions and Formulas
3. Use formulas from science to solve application
problems
e.g.7 LAND of ENCHANTMENT The record high
temperature for New Mexico is 122°F, on June 27,
1994. Convert this temperature to degree Celsius.
Ans. 50°C
Celsius-to-Fahrenheit:
Fahrenheit-to-Celsius:
F = 95 C + 32
C=
5 ( F − 32 )
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3.2 Evaluating Algebraic Expressions and Formulas
4. Find the mean (average) of a set of values
e.g.9 STUDENT PARKING As part of a traffic survey,
the number of cars in a college parking lot at noon
were counted. The results are listed below.
Mon: 121, Tues: 204, Wed: 166, Thurs: 152, Fri: 87.
Find the mean (average) number of cars in the
parking lot at that time.
Ans. 146 cars
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3.3 Simplifying Algebraic Expressions and the
Distribution Property
1. Simplify products
2. Use the distribution property
3. Distribute a factor of –1
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3.3 Simplifying Algebraic Expressions and the
Distribution Property
1. Simplify products
e.g.1 Simplify:
a.
6·9m
b.
–11a(–10)
e.g.2 Simplify:
a.
–6x ·3y
b.
3(–5b)(2a)
Ans.
54m
Ans.
Ans.
Ans.
110a
–18xy
–30ab
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3.3 Simplifying Algebraic Expressions and the
Distribution Property
2. Use the distribution property
The Distributive Property
For any numbers a, b and c
a(b + c) = ab + ac and
e.g.3 Multiply:
a.
10(d + 7)
b.
4(3y – 8)
a(b – c) = ab – ac
Ans.
10d + 70
Ans.
12y – 32
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3.3 Simplifying Algebraic Expressions and the
Distribution Property
2. Use the distribution property
e.g.4
a.
b.
c.
d.
Multiply:
–2(21x + 3)
–8(4 – 10n)
–4(–3t – 16)
–1(a – 7)
Ans.
−42x − 6
Ans.
−32 + 80n
Ans.
12t + 64
Ans.
−a + 7
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3.3 Simplifying Algebraic Expressions and the
Distribution Property
2. Use the distribution property
Distributive property can be extended, for example,
(b + c)a = ba + ca,
(b – c)a = ba – ca
It can also be extended to more then two terms:
a(b + c + d) = ab + ac + ad
a(b – c – d) = ab – ac – ad
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3.3 Simplifying Algebraic Expressions and the
Distribution Property
2. Use the distribution property
e.g.5
a.
b.
c.
d.
Multiply:
(14x + 21)3
(32 – n)4
3(x – 6y)2
–7(–2a – 3b +1)
Ans.
42x + 63
Ans.
128 − 4n
Ans.
6x − 36y
Ans.
14a + 21b − 7
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3.3 Simplifying Algebraic Expressions and the
Distribution Property
3. Distribution a factor of –1
The opposite of a Sum
The opposite of a sum is the sum of opposite
For any numbers a and b,
–(a + b) = –a + (–b)
e.g.6 Simplify:
–(16r – 8)
Ans.
−16r + 8
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3.4 Combining Like Terms (long)
1. Identifying terms and coefficients of terms
2. Identify like terms
3. Combine like terms
4. Find perimeter of a rectangle and square
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3.4 Combining Like Terms
1. Identifying terms and coefficients of terms
Addition symbols separates expressions into parts called
Terms. For example, x + 8 has two terms:
x
+
8
1st term
2nd term
Since subtraction can be written as addition of opposite,
the expression a2 – 3a – 9 has three terms:
a2 – 3a – 9 = a2 +
(–3a)
+
(–9)
1st term
2nd term
3rd term
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3.4 Combining Like Terms
1. Identifying terms and coefficients of terms
In general, a term is a product or quotient of numbers
and/or variables. A single number or variable is also a
term. Examples of terms are:
3
3
5
2
4,
y,
6r, –w ,
7x ,
,
–15ab
n
e.g.1 Identifying terms of each expression:
Ans. three terms: 2y2, 7y, 15
a.
2y2 + 7y + 15
b.
–6ab
Ans. one term: –6ab
Ans. four terms: t, –4, –9t, 8
c.
t – 4 – 9t + 8
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3.4 Combining Like Terms
1. Identifying terms and coefficients of terms
It is important to distinguish between terms and factors.
e.g.2 Is x used as a factor or a term in each expression?
Ans. term
a.
x+2
b.
2x
Ans. factor
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3.4 Combining Like Terms
1. Identifying terms and coefficients of terms
Coefficient: the numerical factor of a term is called the
coefficient of the term.
e.g.3 Identify the coefficients of each term in the
Expression: p3 – 12p2 + 3p – 4
Ans. 1, –12, 3, –4
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3.4 Combining Like Terms
2. Identify like terms
Like terms
Like terms are terms containing exactly the same variables
raised to the same powers.
Any two constants are like terms. For example, in
expression x – 4 – 9x + 8, –4 and 8 are like terms.
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3.4 Combining Like Terms
2. Identify like terms
e.g.4 Identify the like terms in each expressions:
Ans. 6x and 2x are like terms
a.
6x + 7 + 2x
Ans. no like terms
b.
8a4 – 8a3 – 8a2
c.
–4t3 + 1 – 9 + t3 Ans. –4t3 and t3 are like terms,
1 and –9 are like terms
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3.4 Combining Like Terms
3. Combine like terms
3x + 4x can be simplified (why)
3x + 4y cannot be simplified
Combining Like Terms
Like terms can be combined by adding or subtracting the
coefficients of the terms and keeping the same variables
with the same exponents
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3.4 Combining Like Terms
3. Combine like terms
e.g.5 Simplify by combining like terms, if possible:
Ans. 8x
a.
6x + 2x
Ans. –11m
b.
–7m + (–6m) + 3m
Ans. 3t3
c.
7t3 – 4t3
Ans. does not simplify
d.
9r + 11
Ans. 12t
e.
6t + 9t – 3t
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3.4 Combining Like Terms
3. Combine like terms
e.g.6 Simplify by combining like terms:
Ans. y
a.
12y – 11y
Ans. 11y
b.
12y – y
Ans. −y
c.
11y – 12y
Ans. 13y
d.
12y + y
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3.4 Combining Like Terms
3. Combine like terms
e.g.7 Simplify:
5x2 + 13x – 3x – 4
Ans. 5x2 + 10x – 4
e.g.8 Simplify:
7(b + 6) – 1 – (3b – 9)
Ans. 4b + 50
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3.4 Combining Like Terms
4. Finding the perimeter of a rectangle and square
l
Rectangle
w
w
l
The Formula for the Perimeter of a Rectangle
The perimeter P of a rectangle with length l and width w is
given by
P = 2l + 2w
Why?
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3.4 Combining Like Terms
4. Finding the perimeter of a rectangle and square
s
Square
s
s
s
The Formula for the Perimeter of a Rectangle
The perimeter P of a rectangle with sides of length s is
given by
P = 4s
Why?
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3.5 Simplifying Expressions to Solve Equations
1. Determine whether a number is a solution
2. Combine like terms to solve equations
3. Solve equations that have variable terms on both sides
4. Use the distributive property to solve equations
5. Applying strategies to solve equations
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3.5 Simplifying Expressions to Solve Equations
1. Determine whether a number is a solution
e.g.1 Is 8 a solution of
2x – 4 = x + 3?
Ans. no
---------------------------------------------------------------------2. Combine like terms to solve equations
Ans. m = 6
e.g.2 Solve:
10m – 6m = 24
Ans. a = 18
e.g.3 Solve:
100 + 27 = 8a – 35 + a
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3.5 Simplifying Expressions to Solve Equations
3. Solve equations that have variable terms on both sides
e.g.4 Solve:
e.g.5 Solve:
Ans. x = –26
7x – 22 = 8x + 4
–38 – 2b – b = 4b + 11 Ans. x = –7
---------------------------------------------------------------------4. Use the distributive property to solve equations
e.g.6 Solve:
4(8n + 7) =28
Ans. n = 0
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3.5 Simplifying Expressions to Solve Equations
5. Apply a strategy to solve equations
Strategy for Solving Equations (for detail read the book)
1) simplify each side of equation
2) isolate the variable term on one side
3) isolate the variable
4) Check the result
e.g.7 Solve:
3(5x – 40) + 9x = 6(x + 70) Ans. x = 30
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3.6 Using Equations to Solve Application Problems
1. Solve application problems to find one unknown
2. Solve application problems to find two unknowns
3. Solve number-valued problems
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3.6 Using Equations to Solve Application Problems
1. Solve application problems to find one unknown
e.g.1 COLLECTORS A poster collector purchases 6
classic film posters each year. If he now owns 36
posters, in how many years will he have a collection
of 120 posters?
Ans. In 14 years he will have a collection of 120 posters.
Five-step problem solving strategy
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3.6 Using Equations to Solve Application Problems
1. Solve application problems to find one unknown
e.g.2 TAX REFUNDS After receiving tax refund check, a
husband and wife split the refunded money equally.
The wife then give $110 of her share to her son,
leaving her with $344. What was the amount of the
tax refund check?
Ans. The amount of the tax refund check is $908.
Five-step problem solving strategy
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3.6 Using Equations to Solve Application Problems
2. Solve application problems to find two unknowns
e.g.3 MARCHING BAND At a field competition, a
marching band scored 6 points more on the music
portion of the judging than they did on the visual
portion. If their combined score was 84, what were
their scores on music portion and on the visual
portion?
Ans. The amount of the tax refund check is $908.
•
•
Five-step problem solving strategy
Two unknown, one variable
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3.6 Using Equations to Solve Application Problems
2. Solve application problems to find two unknowns
e.g.4 GEOMETRY The perimeter of a rectangle is 100
inches. Find the length and width if the length is four
times the width.
Ans. The width is 10 inches,
the length is 40 inches.
•
•
Five-step problem solving strategy
Two unknown, one variable
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3.6 Using Equations to Solve Application Problems
3. Solve number-valued problems
Number-Value Formula
(Number)(Value) = Total Value
e.g.5 Bananas sells for 99 cents a pound. Find the cost of:
a.
6 lb of bananas Ans. the cost is 994 cents or $9.94
b.
p lb of bananas Ans. the cost is 99p cents.
c.
(p + 3) lb of bananas Ans. the cost is 99(p + 3) cents.47
3.6 Using Equations to Solve Application Problems
3. Solve number-valued problems
e.g.6 TICKET SALES Nine hundred fifty tickets were
sold for a concert. Ticket price were $55 for general
admission and $105 for reserved. Write algebraic
expressions that represent the income received from
the sale of general admission and the sale of reserved
tickets. Use a table to present the result.
Ans. let g be the number of tickets sold for general admission,
then, the income from the sale of general admission = 55g dollars,
The income from the sale of reserved tickets = 105(950 – g) dollars
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