Transcript Document

MM150 – Unit 3 Seminar
• Wednesday, September 7, 2011
• 7 PM ET
• Professor Jack Refling
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Definitions
Algebra: a generalized form of arithmetic.
Variable: symbol used to represent a number
Constant: symbol used to represent a specific
quantity
Algebraic Expression: a collection of variables,
numbers, parentheses, & operation symbols.
Examples: x, x  4, 4(3 y  5), 4 x  2 , y 2  8 y  2
3x  5
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Order of Operations
First do what is “inside Parentheses”, or other grouping
symbols, ( ), [ ], { }, | |, /
If / work top & bottom separately & do the division
last.
Then work with the Exponential parts or Roots, ^ or √
Multiplication and Division are worked next from LEFT to
RIGHT!
Lastly Addition and Subtraction are performed, again from
LEFT to RIGHT!
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Example: Substituting for Two
Variables
Evaluate :
– 4x2 + 3xy – 5y2
Solution:
– 4x2 + 3xy – 5y2
– 4(-3)2 + 3(-3)(4) – 5(4)2
– 4(-3)2 + 3(-3)(4) – 5(4)2
– 4*9 + 3*(-12) – 5*16
– 36 + –36 – 80
– 72 – 80
– 152
when x = - 3 and y = 4
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Definitions
• 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.
2x, 7
 5x 3 , 6x 2
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Properties of the Real Numbers
a(b + c) = ab + ac
a+b=b+a
ab = ba
(a + b) + c = a + (b + c)
(ab)c = a(bc)
Distributive property
Commutative property of
addition
Commutative property of
multiplication
Associative property of
addition
Associative property of
multiplication
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Example: Combine Like Terms
a) 8x + 4x
= (8 + 4)x
= 12x
c) x + 15 - 5x + 9
= 1x - 5x + 15 + 9
= (1- 5)x + (15 + 9)
= -4x + 24
b) 5y - 6y
= (5 - 6)y
= -y
d) 3x + 2 + 6y - 4 + 7x
= (3 + 7)x + 6y + (2 - 4)
= 10x + 6y - 2
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Solving Equations
Addition Property of Equality
If a = b, then a + c = b + c
for all real numbers a, b, and c.
Find the solution to the equation
x - 9 = 24
Check: x - 9 = 24
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Solving Equations continued
Subtraction Property of Equality
If a = b, then a - c = b - c for all real numbers a, b,
and c.
Find the solution to the equation
x + 12 = 31
Check: x + 12 = 31
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Solving Equations continued
Multiplication Property of Equality
If a = b, then a * c = b * c for all real numbers a, b,
and c, where c ≠ 0.
Find the solution to the equation
x
 9.
7
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Solving Equations continued
Division Property of Equality
If a = b, then a b
for all real numbers a, b,

and c, c ≠ 0.
c c
Find the solution to the equation 4x = 48.
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General Procedure for Solving
Linear Equations
1) If an equation involves a fraction or a decimal, remove them
first
2) Use the Distributive Property to remove parentheses.
3) Simplify each side of the equation, by combining like terms.
4) Separate variable & constant terms - use the Addition
Property
5) Move the coefficient of the variable using the Multiplication
Property.
6) Check solution by substituting back into the original equation.
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Example: Solving Equations
Solve 3x - 4 = 17
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Example: Solving Equations
Solve 8x + 3 = 6x + 21
8 x  3  6 x  21
8 x  3  3  6 x  21  3
8 x  6 x  18
8 x  6 x  6 x  6 x  18
2 x  18
2 x 18

2
2
x 9
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Proportions
• A proportion is a statement of equality
between two ratios. a  c
b
d
• Cross Multiplication
If a  c , then ad = bc, b ≠ 0, d ≠ 0.
b
d
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Example
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
754 pounds of fertilizer
753.33  x
would be needed.
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Definitions
• A formula is an equation that typically has a
real-life application.
• To evaluate a formula, substitute the given
value for their respective variables and then
evaluate using the order of operations.
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Perimeter
The formula for the perimeter of a rectangle is
Perimeter = 2 * length + 2 * width or P = 2l + 2w.
Use the formula to find the perimeter of a yard
when l = 150 feet and w = 100 feet.
P = 2l + 2w
P = 2(150) + 2(100)
P = 300 + 200
P = 500 feet
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Example
The formula for the volume of a cylinder is
V = r2h. Use the formula to find the height of a
cylinder with a radius of 6 inches and a volume
of 565.49 in3.
V   r 2h
565.49   (62 )h
565.49  36 h
565.49 36 h

36
36
5.000  h
The height of the cylinder is 5 inches.
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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
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Translating Words to Expressions
Phrase
Mathematical
Expression
Ten more than a number
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
Matching Game & more translations on
http://www.ramshillfarm.com/Math/Math150/Unit_3.html
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Translating Words to Expressions
Phrase
Four times a number
2 decreased by a number
Mathematical
Expression
4x
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
Seven more than a number is 12
Mathematical Equation
x + 7 = 12
Three less than a number is 4
x-3=4
Twice a number, decreased by 3
is 8.
2x - 3 = 8
A number decreased by 15 is 9
times the number
x - 15 = 9x
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To Solve a Word Problem
1. Read the problem carefully at least twice to be sure
that you understand it.
a) If possible, draw a sketch to help visualize the problem.
2. Determine which quantity you are being asked to find.
a) Choose a letter to represent this unknown quantity. Write
down exactly what this letter represents.
3.
4.
5.
6.
Write the word problem as an equation.
Solve the equation for the unknown quantity.
Answer the question or questions asked.
Check the solution.
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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?
Let h = the number of hours billed
Cost of parts + labor = total amount
339 + 45h = 496.50
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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.
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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?
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continued, 184 feet of fencing,
length 8 feet longer than width
Let x = width of region
Let x + 8 = length
x
P = 2l + 2w
x+8
184 = 2(x + 8) + 2(x)
184 = 2x + 16 + 2x
42 = x
184 = 4x + 16
-16
- 16
168 = 4x
The width of the region is 42
feet and the length is 50 feet.
168 = 4x
4
4
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