Transcript Document

MM150 Unit 3 Seminar
Sections 3.1 - 3.4
1
3.1
Order of Operations
2
Definitions
• Algebra: a generalized form of arithmetic.
• Variables: letters used to represent numbers
• Constant: a number on it’s own or a symbol/letter that
represents a fixed quantity.
• Algebraic expression: a collection of variables, numbers,
parentheses, and/or operation symbols (+ or x).
Expressions DO NOT have equal signs, “=“.
Examples:
x, x  4, 4(3 y  5),
4x  2
, y 2  8y  2
3x  5
• Algebraic equation: is an algebraic expression that has an equal
sign, “=“.
Examples: 2 + x = 11
3y – 9 = 36
3
Evaluating Expressions
• Exponents:
x2
34
-7y3
59
 2 • 2 • 2 • 2 • 2 • 2 • 2, you can rewrite this as 27
 x • x • x • x = x4
 (2a)(2a)(2a) = (2a)3
 (x + 6)(x + 6) = (x + 6)2
• x^2 is the same as x2
2^3 = 23 = 2*2*2 = 8
Be careful!
(-2)4 = (-2)(-2)(-2)(-2) = 16
-24 = -(2*2*2*2) = -16
(-2)4 is not equal to -24
4
Order of Operations
1. Parentheses: Perform all operations inside
parentheses or other grouping symbols (use the rules
below).
2. Exponents: perform all exponential operations or
find any roots.
3. Multiplication/Division: perform all multiplication or
division whichever comes first from left to right.
4. Addition/Subtraction: perform all addition or
subtraction whichever comes first from left to right.
Please Excuse My Dear Aunt Sally
Perform the one that comes
first from left to right
PEMDAS
Perform the one that comes
first from left to right
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Example of Evaluating
an Expression
Evaluate the expression x2 + 4x + 5
when x = 3.
• Solution:
x2 + 4x + 5
= 32 + 4(3) + 5
= 9 + 12 + 5
Be sure to
follow the
Order of
Operations!
= 26
6
Example of Substituting
for Two Variables
Evaluate 4x 2  3xy  5y 2
when x = 3 and y = 4.
• Solution: 4 x 2  3 xy  5y 2
 4(3)2  3(3)(4)  5(4 2 )
 4(9)  36  5(16)
 36  36  80
 0  80
 80
Be sure to
follow the
Order of
Operations!
7
Examples of Checking Solutions
A. Determine if 9 is the solution to 2 + x = 11.
We can check by substituting 9 for x.
2 + x = 11
2 + 9 = 11
11 = 11
This is a true statement, therefore 9 is
a solution to 2 + x = 11
B. Determine if 10 a solution to 3y - 9 = 36.
We can check by substituting 10 for y.
3y - 9 = 36
3(10) - 9 = 36
30 - 9 = 36
21 =/= 36
This is a false statement, therefore 10
is NOT a solution to 3y – 9 = 36
8
EVERYONE:
page 111 #42
Cost of a Tour: The cost, in dollars, for Crescent City
Tours to provide a tour for x people can be determined by
the expression 220 + 2.75x. Determine the cost for
Crescent City Tours to provide a tour for 75 people.
Cost = 220 + 2.75x
Substitute
Cost = 220 + 2.75(75)
Multiply
Cost = 220 + 206.25
Add
Cost = 426.25
$426.25
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3.2
Linear Equations in
One Variable
10
Terms
• Terms: parts that are added or subtracted in an algebraic
expression
• Terms can be:
Coefficient: numerical part of the term.
 Constants: 3, - 5, 0, ⅜, 
Example: in the term 3x, 3 is the coefficient
 Variables: a, b, c, x, y, z
Example: in - 99ay5, - 99 is the coefficient
 Products: 3x, ab2, - 99ay5
• Expressions can have:
 one term (monomial):
x
5t
 two terms (binomial):
y+9
- 6s - 11
 three terms (trinomial):
- 10y
x2 + 7x - 10
 four terms or more (polynomial):
NOTE:
Decreasing power
of the variable.
x2y + xy - 11y + 23
11
Like and Unlike Terms
• Like Terms: have the same variables with same
exponents on the variables.




5x and 3x are like terms
6ab and -9ab are like terms
16x2 and x2 are like terms
-0.35ac5 and -400ac5 are like terms
• Unlike Terms: have different variables or different
exponents on the variables.




5x and 3 are unlike terms
6b and -9c are unlike terms
16x and x2 are unlike terms
-0.35a5c and -400ac5 are unlike terms
12
Example: Combine Like Terms
• 8x + 4x
= (8 + 4)x
= 12x
•
5y - 6y
= (5 - 6)y
= -y
• 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
Add or subtract the coefficients of like
terms and KEEP the same variable part
13
EVERYONE:
page 113 #32
Combine like terms: 6(r - 3) - 2(r + 5) + 10
6(r - 3) - 2(r + 5) + 10
Distribute
= 6r - 18 - 2r - 10 + 10
Combine like terms
= 4r – 18
Finished!
Note that 4r and - 18 are unlike terms,
therefore you cannot combine them
14
Addition Property of Equality
For real numbers a, b, and c
if a = b, then a + c = b + c.
• Example: Solve x - 9 = 15
x - 9 = 15
x - 9 + 9 = 15 + 9
x + 0 = 24
x = 24
addition property
15
Subtraction Property of Equality
For real numbers a, b, and c
if a = b, then a - c = b – c.
• Example: Solve x + 11 = 19
x + 11 = 19
x + 11 - 11 = 19 - 11
x=8
subtraction property
16
Multiplication Property of Equality
For real numbers a, b, and c, where c =/= 0
if a = b, then a • c = b • c
Example: Solve
x
 9.
7
x
9
7
 x
7     7(9)
 7
1
7x
1
 63
7
x  63
Multiplication Property
17
Division Property of Equality
For real numbers a, b, and c, where c =/= 0
a b
if a = b, then 
c c
Example: Solve 4x = 48
4 x  48
4 x 48

4
4
x  12
Division Property
18
Steps for Solving Equations
1. Simplify (clean up) both sides of the equation by:
a.) Get rid of any fractions by multiplying both sides of the
equation by the LCD.
b.) Use the distributive property to get rid of parentheses
when necessary.
c.) Combine like terms on same side of equal sign when
possible.
Equation will be in the form ax + b = cx + d
2. Collect all the variables on one side of the equal sign and all
constants to the other side by using the addition/subtraction
property.
Equation will be in the form ax = b
3. Solve for the variable using the division/multiplication
property.
The resulting form will be x = c
19
EVERYONE: Solve for x
3x - 4 = 17
3 x  4  17
3 x  4  4  17  4
3 x  21
3 x 21

3
3
x 7
20
EVERYONE: Solve for x
21 = 6 + 3(x + 2)
21  6  3( x  2)
21  6  3 x  6
21  3 x  12
21  12  3 x  12  12
9  3x
9 3x

3 3
3x
21
EVERYONE: Solve for x
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
22
Proportions
• A proportion is a statement of equality
between two ratios.
a c

b d
• Cross Multiplication
a c
 , then ad = bc,
If
b d
a c

b d
b =/= 0, d =/= 0.
a•d
b•c
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To Solve Application Problems
Using Proportions
1. Represent the unknown quantity by a variable.
2. Set up the proportion by listing the given ratio on the
left-hand side of the equal sign and the unknown and
other given quantity on the right-hand side of the
equal sign.
When setting up the proportion, the same
respective quantities should occupy the same
respective positions on the left and right.
For example, an acceptable proportion might be
miles miles

hour
hour
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To Solve Application Problems
Using Proportions (continued)
3. Once the proportion is properly
written, drop the units and use cross
multiplication to solve the equation.
4. Answer the question or questions
asked using appropriate units.
25
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
753.33  x
754 pounds of
fertilizer would
be needed.
26
3.3
Formulas
27
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
P
P
P
= 2L + 2W
= 2(150) + 2(100)
= 300 + 200
= 500 feet
28
Volume of a Cylinder
• The formula for the volume of a cylinder is
V = (pi)(r2)(h).
• 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.
29
EVERYONE: Solve the equation for y
-9x + 4y = 11
-9x + 4y = 11
9x - 9x + 4y = 9x + 11
4y = 9x + 11
30
EVERYONE: Solve the equation for y
5x + 3y - 2z = 22
5x + 3y - 2z = 22
-5x + 5x + 3y - 2z = -5x + 22
3y - 2z = -5x + 22
3y - 2z + 2z = -5x + 22 + 2z
3y = -5x + 2z + 22
31
EVERYONE: Solve the equation for y
3x + 8y - 9 = 0
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
32
Solve
h
A  (b1  b2 )
2
for b2.
h
A  (b1  b2 )
2
h

2   A   2   (b1  b2 ) 
2

2 A  h(b1  b2 )
2 A h(b1  b2 )

h
h
2A
 b1  b2
h
2A
 b1  b2
h
33
3.4
Applications of Linear
Equations in One
Variable
34
Translating to Math
•
Six more than a number
6+x
•
A number increased by 3
x+3
•
Four less than a number
x–4
•
Twice a number
2x
•
Four times a number
4x
•
3 decreased by a number
3–x
•
The difference between a number and 5
x–5
•
Four less than 3 times a number
3x – 4
•
Ten more than twice a number
2x + 10
•
The sum of 5 times a number and 3
5x + 3
•
Eight times a number, decreased by 7
8x – 7
•
Six more than a number is 10
x + 6 = 10
•
Five less than a number is 20
x – 5 = 20
•
Twice a number, decreased by 6 is 12
2x – 6 = 12
•
A number decreased by 13 is 6 times the number
x – 13 = 6x
35
To Solve a Word Problem
1. Read the problem carefully at least twice to be sure
that you understand it.
2. If possible, draw a sketch to help visualize the
problem.
3. Determine which quantity you are being asked to
find. Choose a letter to represent this unknown
quantity. Write down exactly what this letter
represents.
4. Write the word problem as an equation.
5. Solve the equation for the unknown quantity.
6. Answer the question or questions asked.
7. Check the solution.
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
37
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.
38
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?
39
Example continued
184 feet of fencing, length 8 feet longer than width
• Let x = width of region
• Let x + 8 = length
• P = 2L + 2W
184  2( x )  2( x  8)
184  2 x  2 x  16
184  4 x  16
168  4 x
42  x
x
x+8
The width of the
region is 42 feet
and the length is
50 feet.
40
Page 139 #34
• PetSmart has a sale offering 10% off of all
pet supplies. If Amanda spent $15.72 on pet
supplies before tax, what was the price of
the pet supplies before the discount?
• the price before discount will be called “x”.
x - x (0.10) = 15.72
x - 0.10x = 15.72
0.9x = 15.72
x ≈17.47
The price is
≈ $17.47
41
Page 140 #46
•
A bookcase with three shelves is built by a student. If the height of
the bookcase is to be 2 ft longer than the length of a shelf and the
total amount of wood to be used is 32 ft, find the dimensions of the
bookcase.
•
Let x = width (length of shelf) and let x + 2 = height
•
From picture in book, there are 4 pieces of wood for width and 2 pieces
of wood for the height.
•
4(width) + 2(height) = total amount of wood
4x + 2(x + 2) = 32
4x + 2x + 4 = 32
6x + 4 = 32
6x = 28
So, width of bookcase is
and height is
ft
x=
42
ft