Transcript Document

Welcome to MM150!
Kirsten K. Meymaris
Unit 3
Plan for the hour
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Order of Operations (3.1) Learning the Rules
Linear Equations, One Variable (3.2) Setting up the Game
Formulas (3.3) Playing the Game
Applications of Linear Equations,
One Variable (3.4) New Games
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3.1
Order of Operations
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Order of Operations
1. All operations within Parentheses or other
grouping symbols (according to the following
order).
2. All Exponential operations (raising to powers
or finding roots).
3. All Multiplication and Divisions - left to right.
4. Additions and Subtractions - left to right.
PEMDAS
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Order of Operations
1. All operations within Parentheses or other
grouping symbols (according to the following
order).
2. All Exponential operations (raising to powers
or finding roots).
3. All Multiplication and Divisions - left to right.
4. Additions and Subtractions - left to right.
PEMDAS, Also known as:
“Please Excuse My Dear Aunt Sally”
“Place Everything Math Down Another Subway”
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example: Evaluating an Expression
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Evaluate the expression x2 + 4x + 5 for x = 3.
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Example: Evaluating an Expression
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Evaluate the expression x2 + 4x + 5 for x = 3.
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Solution:
x2 + 4x + 5
= 32 + 4(3) + 5
= 9 + 12 + 5
= 26
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Example: Evaluating an Expression
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Evaluate the expression: 7x2 + 5x – 11 for x = -1
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Example: Evaluating an Expression
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Evaluate the expression: 7x2 + 5x – 11 for x = -1
Solution:
= (7)(-1)2 + (5)(-1) – 11
= (7)(1) + (5)(-1) – 11
= 7 + (-5) – 11
= 7 – 5 – 11
= -9
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3.2
Linear Equations in One Variable
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Definitions
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Terms are parts that are added or subtracted in
an algebraic expression.
Coefficient is the numerical part of a term.
Like terms are terms that have the same
variables with the same exponents on the
variables.
2x, 7x
 5x 2 , 8x 2
Unlike terms have different variables or different
exponents on the variables.
3
2
2x, 7
 5x , 6x
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Properties of the Real Numbers
a(b + c) = ab + ac
a+b=b+a
Distributive property
Commutative property
of addition
ab = ba
Commutative property
of multiplication
(a + b) + c = a + (b + c) Associative property of
addition
(ab)c = a(bc)
Associative property of
multiplication
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Example: Combine Like Terms
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8x + 4x
5y  6y
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x + 15  5x + 9
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3x + 2 + 6y  4 + 7x
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Example: Combine Like Terms
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8x + 4x
= (8 + 4)x
= 12x
5y  6y
= (5  6)y
= y
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x + 15  5x + 9
= (1 5)x + (15 + 9)
= 4x + 24
3x + 2 + 6y  4 + 7x
= (3 + 7)x + 6y + (2  4)
= 10x + 6y  2
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Properties of Equality
Addition Property of Equality
If a = b, then a + c = b + c for all real numbers a, b, and c.
Subtraction Property of Equality
If a = b, then a  c = b  c for all real numbers a, b, and c.
Multiplication Property of Equality
If a = b, then a • c = b • c for all real numbers a, b, and c, where c  0.
Division Property of Equality
If a = b, then
a b

c c
for all real numbers a, b, and c, c  0.
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
General Procedure for Solving Linear
Equations
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Eliminate fractions with LCD (or LCM)
Remove parentheses when necessary
Combine like terms
Collect all variables on one side, all constants
on other
Solve
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Example: Solving Equations
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Solve 3x  4 = 17.
•Eliminate fractions with LCD (or LCM)
•Remove parentheses when necessary
•Combine like terms
•Collect all variables on one side, all constants on other
•Solve
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Example: Solving Equations
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Solve 3x  4 = 17.
3 x  4  17
3 x  4  4  17  4
3 x  21
3 x 21

3
3
x 7
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Example: Solving Equations
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Solve 8x + 3 = 6x + 21.
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Example: Solving Equations
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Solve 8x + 3 = 6x + 21.
8 x  3  6 x  21
8 x  3  3  6 x  21  3
8x
8x  6x
2x
2x
2
x
 6 x  18
 6 x  6 x  18
 18
18

2
9
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Example: Solving Equations
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Solve 6(x  2)  2x  3  4(2x  3)  2.
6( x  2)  2 x  3  4(2 x  3)  2
6 x  12  2 x  3  8 x  12  2
8 x  9  8 x  10
8 x  8 x  9  8 x  8 x  10
9  10
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False, the equation has no solution. The
equation is inconsistent.
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Example: Solving Equations
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Solve 4(x  1)  6(x  2)  2(x  4).
4( x  1)  6( x  2)  2( x  4)
4 x  4  6 x  12  2 x  8
2 x  8  2 x  8
2 x  2 x  8  2 x  2 x  8
8  8
8  8  8  8
00
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True, 0 = 0 the solution is all real numbers.
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Proportions
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A proportion is a statement of equality between
two ratios.
a c

b d
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Cross Multiplication
a c
If  , then ad = bc, b  0, d  0.
b d
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Example
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If the ratio of boys to girls is 1:3 and there are 213
girls present, how many boys are present?
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Example
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If the ratio of boys to girls is 1:3 and there are 213
girls present, how many boys are present?
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Example

If my Subaru can go 315 miles on about 1 tank of
gas (13 gallons of gas), how many gallons of gas
are needed for my trip from Atlanta to New York
City (850 miles)?
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Example

If my Subaru can go 315 miles on about 1 tank of
gas (13 gallons of gas), how many gallons of gas
are needed for my trip from Atlanta to New York
City (850 miles)?
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Example
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A 50 pound bag of fertilizer will cover an area of
15,000 ft2. How many pounds are needed to
cover an area of 226,000 ft2?
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Example
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A 50 pound bag of fertilizer will cover an area of
15,000 ft2. How many pounds are needed to
cover an area of 226,000 ft2?
50 pounds
x

2
15,000 ft
226,000 ft 2
(50)(226,000)  15,000 x
11,300,000  15,000 x
11,300,000 15,000 x

15,000
15,000
753.33  x
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754 pounds of fertilizer
would be needed.
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Examples – Open Forum!
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3.3
Formulas
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Exponential Equations: Carbon Dating
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Carbon dating is used by scientists to find the
age of fossils, bones, and other items. The
 t / 5600
formula used in carbon dating is P  P0 2
.
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example
If 15 mg of C14 is present in an animal bone
recently excavated, how many milligrams will be
present in 4000 years?
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Example
If 15 mg of C14 is present in an animal bone
recently excavated, how many milligrams will be
present in 4000 years?
P0 = original amount =
P= amount after t number of years =
t = number of years =
P = P02-t/5600
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example
If 15 mg of C14 is present in an animal bone
recently excavated, how many milligrams will be
present in 4000 years?
P0 = original amount = 15 mg
P= amount after t number of years = ?
t = number of years = 4000
P = P02-t/5600
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Exponential Equations: Carbon Dating
continued
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P = 15(2)-4000/5600
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Example: Carbon Dating
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P  P0 2
 t / 5600
P  15(2)4000 / 5600
P  15(2).71
P  15(0.61)
P  9.2 mg
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In 4000 years, approximately 9.2 mg of the
original 15 mg of C14 will remain.
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Solving for a Variable in a
Formula or Equation
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Solve the equation 3x + 8y  9 = 0 for y.
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Solving for a Variable in a
Formula or Equation

Solve the equation 3x + 8y  9 = 0 for y.
3 x  8y  9  0
3 x  8y  9  9  0  9
3 x  8y  9
3 x  3 x  8y  9  3 x
8y  9  3 x
8y 9  3 x

8
8
9  3x
y
8
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
3.4
Applications of Linear
Equations in One Variable
(Word Problems!)
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Translating Words to Expressions
Phrase
Mathematical
Expression
Ten more than a number
A number increased by 5
Four less than a number
A number decreased by
8
Twice a number
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Translating Words to Expressions
Phrase
Ten more than a number
Mathematical
Expression
x + 10
A number increased by 5
x+5
Four less than a number
x–4
A number decreased by
8
x–8
Twice a number
2x
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Translating Words to Expressions
Phrase
Mathematical
Expression
Four times a number
2 decreased by a number
The difference between a
number and 6
Five less than 7 times a
number
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Translating Words to Expressions
Phrase
Four times a number
Mathematical
Expression
4x
2 decreased by a number
2–x
The difference between a
number and 6
x–6
Five less than 7 times a
number
7x – 5
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Translating Words to Expressions
Phrase
Mathematical
Equation
Seven more than a
number is 12
Three less than a
number is 4
Twice a number,
decreased by 3 is 8.
A number decreased by
15 is 9 times the number
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Translating Words to Expressions
Phrase
Seven more than a
number is 12
Three less than a
number is 4
Twice a number,
decreased by 3 is 8.
A number decreased by
15 is 9 times the number
Mathematical
Equation
x + 7 = 12
x–3=4
2x  3 = 8
x  15 = 9x
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
To Solve a Word Problem
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Read problem three (yes, 3!) times
Sketch it
Write what you know and want to know
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Write out equation
Solve for unknown
Answer the question(s)
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assign variables
Include units
Double check your answer
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Example
The bill (parts and labor) for the repairs of a car
was $496.50. The cost of the parts was $339.
The cost of the labor was $45 per hour. How
many hours were billed?
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Example
The bill (parts and labor) for the repairs of a car
was $496.50. The cost of the parts was $339. The
cost of the labor was $45 per hour. How many
hours were billed?
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h = # of hours billed
parts + labor = total
339 + 45h = 496.50
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example continued
339  45h  496.50
339  339  45h  496.50  339
45h  157.50
45h 157.50

45
45
h  3.5
The car was worked on for 3.5 hours.
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example:
Discussion Board Highlight
The daycare my son attends charges $20.00 per day, 5
days a week. Children must be picked up by 6:00 pm
otherwise they charge $5.00 for every 5 minutes you are
late. I was 5 minutes late 3 times last week. How much
did I pay that week? (Assuming my son went every day
this week.)
X = number of days at daycare =
Y = number days late =
T = total cost each week =
20x + 5y = T
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example:
Discussion Board Highlight
X=5
Y=3
20x + 5y = T
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Example
At Action Water Sport of Ocean, City Maryland,
the cost of renting a Jet ski is $42 per half hour,
which includes a 5% sales tax. Determine the
cost of a half hour Jet Ski rental before tax.
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example
At Action Water Sport of Ocean, City Maryland,
the cost of renting a Jet ski is $42 per half hour,
which includes a 5% sales tax. Determine the
cost of a half hour Jet Ski rental before tax.
R = rental cost = ?
tax = 5% = 0.05
T = total cost including tax = $42
R + (0.05)R = T
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Example – Jet Ski rental
R + (0.05)R = T
R + (0.05)R = 42
R(1+0.05) = 42
R(1.05) = 42
R = 40
The rental cost $40 per half hour before tax.
Double Check:
40+(0.05)40 =? 42
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Example

Sandra Cone wants to fence in a rectangular
region in her backyard for her lambs. She only
has 184 feet of fencing to use for the perimeter
of the region. What should the dimensions of
the region be if she wants the length to be 8 feet
greater than the width?
Adapted from Pearson Education , Inc. Copyright © 2009 Pearson Education, Inc.
Example

Sandra Cone wants to fence in a rectangular
region in her backyard for her lambs. She only
has 184 feet of fencing to use for the perimeter
of the region. What should the dimensions of
the region be if she wants the length to be 8 feet
greater than the width?
x
x+8
x = width of region
x + 8 = length
P = 2l + 2w
184 = 2(x) + 2 (x+8)
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Example - fencing
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x = width of region
x + 8 = length
P = 2l + 2w
x
x+8
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Example - fencing
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x = width of region
x + 8 = length
P = 2l + 2w
x
x+8
184  2( x )  2( x  8)
184  2 x  2 x  16
184  4 x  16
168  4 x
42  x
The width of the region is 42
feet and the length is 50 feet.
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Examples – Open Forum!
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Thank You!
Remember to
Ask, Ask, Ask!
[email protected]
AIM: kkmeymaris
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