Transcript Algebra 1 Review - Marquette University High School

By: Max Kent
For Help
refer to :
http://www.mathtv.com/
http://www.quickmath.com/
If the same number is added to both sides of an equation,
the two sides remain equal. That is if
x = y, then x + z = y + z.
Multiplication Property (of
Equality)
For all real numbers a and b , and for c ≠ 0 ,
a = b is equivalent to ac = bc .
The property that a = a.
Symmetric Property (of Equality)
The following property: If if a = b then b = a.
Transitive Property (of Equality)
The following property: If a = b and b = c, then a = c. One of the equivalence properties
of equality.
Note: This is a property of equality and inequalities. One must be cautious, however,
when attempting to develop arguments using the transitive property in other settings.
Here is an example of an unsound application of the transitive property: "Team A
defeated team B, and team B defeated team C. Therefore, team A will defeat team C.“
http://www.mathwords.com/t/transitive_property.htm
The addition or multiplication of a set of numbers is the same regardless of how the numbers
are grouped. The associative property will involve 3 or more numbers. The parenthesis
indicates the terms that are considered one unit. The groupings (Associative Property) are
within the parenthesis. Hence, the numbers are 'associated' together. In multiplication, the
product is always the same regardless of their grouping. The Associative Property is pretty
basic to computational strategies. Remember, the groupings in the brackets are always done
first, this is part of the order of operations.
When we change the groupings of addends, the sum does not change:
(2 + 5) + 4 = 11 or 2 + (5 + 4) = 11
Associative Property of Multiplication
When three or more numbers are multiplied, the product is the same regardless
of the grouping of the factors.
Example: (2 * 3) * 4 = 2 * (3 * 4)
The Commutative Property of Addition states that changing the
order of addends does not change the sum, i.e. if a and b are two
real numbers, then a + b = b + a.
Commutative Property of Multiplication
The commutative property of multiplication simply means it does
not matter which number is first when you write the
problem. The answer is the same.
3 x 5 = 5 x 3 (The numbers can be switched around and the
answer is the same.)
The distributive property of multiplication over addition is simply this: it makes no
difference whether you add two or more terms together first, and then multiply the
results by a factor, or whether you multiply each term alone by the factor first, and
then add up the results.
That is,
adding up the term first; then multiplying by the factor = multiplying each term
by the factor first, then adding up the resulting terms
That is:
Factor(Term1 + Term2 + ... + TermN) = Factor(Term1) + Factor(Term2) +
..... + Factor(TermN)
If we call the Factor "a," and we call the terms "b", "c,"......"t", then this statement begins to
look like a mathematical statement:
a(b + c + ....... + t) = a(b) + a(c) + .... +a(t)
EXAMPLE: (The factor is 3, and the three terms are 2, 7, -5)
3(2 + 7 - 5) = 3(2) + 3(7) + (3)(-5)
3(4)
= 6 + 21 - 15
12
= 12
When you add a number to its opposite you get zero a+(-a)=0
Prop of Reciprocals or Inverse Prop. of
Multiplication
A reciprocal is the number you have to multiply a given
number by to get 1. Ex) you have to multiply 2 by 1/2 to get
1. therefore the reciprocal of 2 is 1/2
As implied above, a property of two reciprocals is that their
product equals 1.
Another name for "reciprocal" is "multiplicative inverse."
Identity property of addition states that the sum of zero and any
number or variable is the number or variable itself.
For example, 4 + 0 = 4, - 11 + 0 = - 11, y + 0 = y are few examples
illustrating the identity property of addition.
Identity Property of Multiplication
Identity property of multiplication states that the product of 1
and any number or variable is the number or variable itself.
For example, 4 × 1 = 4, - 11 × 1 = - 11, y × 1 = y are few
examples illustrating the identity property of multiplication.
A number times zero equals zero. (7*0=0)
Closure Property of Addition
The closure property of addition says that if you add together any two
numbers from a set, you will get another number from the same set. If
the sum is not a number in the set, then the set is not closed under
addition.
Closure Property of Multiplication
Closure Property:
For any two whole numbers a and b, their product a*b is also a
whole number.
Example: 10*9 = 90
How do you simplify 72 × 76?
If you recall the way exponents are defined, you know that this means:
(7 × 7) × (7 × 7 × 7 × 7 × 7 × 7)
If we remove the parentheses, we have the product of eight 7s, which can be written
more simply as:
78
This suggests a shortcut: all we need to do is add the exponents!
72 × 76 = 7(2 + 6) = 78
In general, for all real numbers a, b, and c,
ab × ac = a(b + c)
To multiply two powers with the same base, add the exponents.
Power of a Product Property
To find a power of a product, find the power of each factor and then multiply. In general,
(ab)m = am · bm.
Power of a Power Property
To find a power of a power, multiply the exponents. This is an extension of the product of
powers property. Suppose you have a number raised to a power, and you multiply the whole
expression by itself over and over. This is the same as raising the expression to a power:
(53)4 = (53)(53)(53)(53)
This property states that to divide powers having the same
base, subtract the exponents.
Power of a Quotient Property
This property states that the power of a quotient can be
obtained by finding the powers of numerator and denominator
and dividing them.
A number to the power of zero equals 1. 2130457040=1
Negative Power Property
When you have a negative exponent on, say, 4, it will be written 4^2 You basically take the reciprocal of it and change the exponent to
a positive one. 4^-2 would be 1/4^2
The Zero Product Property simply states that if ab
= 0, then either a = 0 or b = 0 (or both). A product
of factors is zero if and only if one or more of the
factors is zero.
The product of the square roots is the square root of the
product.
Quotient of Roots Property
For any non-negative (positive or 0) real number a and any
positive real number b: =√a -- √b
Power of a Root Property
Which property?
X+Y=Y+X
Click for answer
Commutative Property (of Addition)
Which property?
(5x+9x)+3x=(5x+3x)+9X
Click for answer
Associative Property of Addition
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With only one inequality sign = 3x<15 | x<5
Conjunction = 3x<15<5x | x<5 and x>3
Disjunction = 2x>8 or 2x<4 | x>4 or x<2
Solving Inequalities
Linear inequalities are also called first degree inequalities, as
the highest power of the variable (or pronumeral) in these
inequalities is 1. E.g. 4x > 20 is an inequality of the first degree,
which is often called a linear inequality.
Many problems can be solved using linear inequalities.We know
that a linear equation with one pronumeral has only one value for
the solution that holds true. For example, the linear equation 6x =
24 is a true statement only when x = 4. However, the linear
inequality 6x > 24 is satisfied when x > 4. So, there are many
values of x which will satisfy the inequality 6x > 24.
Graph: y=x-5?
Solve: y = 3x – 2
y = –x – 6
Y=-x-6
Now solve for Y
3x-2=-x-6
Y=3(2)-2
4x=8
Y=6-2
X=2
Y=4
The answer is (2,4)
Solve: X2+10x+25+y2?
[X2+10x+25] is a PST
(x+5)2 +y2
Now just factor y and put it in with each
binomial
The answer will be: (x+5+y)(x+5+y)
Solve:
f(x)= is another way to write y=
Functions are relations only when every input has a distinct output, so not all
relations are functions but all functions are relations.
Let’s say you had the points (2,3) and (3,4) and you needed to find a linear function that contained
them. This is how you would do that.
3-4 over (divided by) 2-3 (rise over run, Y is rise, X is run)
you would get -1 over -1. This equals 1, which will be the slope. To find y-intercept, substitute:
2=1(3)+b
2=3+b  -1=b
So your final equation is: Y=X-1. You can now graph this.
Graph: x2-6x+5
The x-intercepts are (5,0) and (1,0).
y-intercept:
The y-intercept is (0,5). and
Vertex:
So the vertex is (3, -4).
Simplify:
The "minus" on the 2 says to move the variable; the "minus" on the 6 says that
the 6 is negative. Warning: These two "minus" signs mean entirely different
things, and should not be confused. I have to move the variable; I should not
move the 6.
Answer:
Example:
Solve: You need a 15% acid solution for a certain test, but your supplier only ships a 10% solution and a 30%
solution. Rather than pay the hefty surcharge to have the supplier make a 15% solution, you decide to mix
10% solution with 30% solution, to make your own 15% solution. You need 10 liters of the 15% acid solution.
How many liters of 10% solution and 30% solution should you use?
Let x stand for the number of liters of 10% solution, and let y stand for the number of liters of 30%
solution. (The labeling of variables is, in this case, very important, because "x" and "y" are not at all
suggestive of what they stand for. If we don't label, we won't be able to interpret our answer in the end.)
For mixture problems, it is often very helpful to do a grid:
Solve: A collection of 33 coins, consisting of nickels, dimes, and quarters, has a value of $3.30. If
there are three times as many nickels as quarters, and one-half as many dimes as nickels, how many
coins of each kind are there?
I'll start by picking and defining a variable, and then I'll use translation to convert
this exercise into mathematical expressions.
Nickels are defined in terms of quarters, and dimes are defined in
terms of
nickels, so I'll pick a variable to stand for the number of
quarters, and then
work from there:
number of quarters: q
number of nickels: 3q
number of dimes: (½)(3q) = (3/2)q
There is a total of 33 coins, so:
q + 3q + (3/2)q = 33
4q + (3/2)q = 33
8q + 3q = 66
11q = 66
q=6
Then there are six quarters, and I can work backwards to figure out that there
are 9 dimes and 18 nickels.
Solve: A wallet contains the same number of pennies, nickels, and dimes. The coins total $1.44.
How many of each type of coin does the wallet contain?
Since there is the same number of each type of coin, I can use one
variable
to stand for each:
number of pennies: p
number of nickels: p
number of dimes: p
The value of the coins is the number of cents for each coin times the number
of that type of coin, so:
value of pennies: 1p
value of nickels: 5p
value of dimes: 10p
The total value is $1.44, so I'll add the above, set equal to 144 cents, and
1p + 5p + 10p = 144
16p = 144
p=9
There are nine of each type of coin in the wallet.
Solve: In three more years, Miguel's grandfather will be six times as old as Miguel was last year. When
Miguel's present age is added to his grandfather's present age, the total is 68. How old is each one
now?
This exercise refers not only to their present ages, but also to both their ages last year and their ages in three
years, so labeling will be very important. I will label Miguel's present age as "m" and his grandfather's present
age as "g". Then m + g = 68. Miguel's age "last year" was m – 1. His grandfather's age "in three more years"
will be g + 3. The grandfather's "age three years from now" is six times Miguel's "age last year" or, in math:
g + 3 = 6(m – 1)
This gives me two equations with two variables:
m + g = 68
g + 3 = 6(m – 1)
Solving the first equation, I get m = 68 – g. (Note: It's okay to solve for "g = 68 – m", too. The problem will
work out a bit differently in the middle, but the answer will be the same at the end.) I'll plug "68 – g" into the
second equation in place of "m":
g + 3 = 6m – 6
g + 3 = 6(68 – g) – 6
g + 3 = 408 – 6g – 6
g + 3 = 402 – 6g
g + 6g = 402 – 3
7g = 399
g = 57
Since "g" stands for the grandfather's current age, then the grandfather is 57 years old. Since m + g = 68,
then m = 11, and Miguel is presently eleven years old.
Example: y = a + bx
x is the independent or predictor variable
and y is the dependent or response
variable. To find a and b we follow the
steps:
The sum of the x-- Sx
The sum of the y-- Sy
The sum of the squares
of x-- Sx2
The sum of the products of
x and y-- Sxy
Used for: Linear regression attempts to
model the relationship between
two variables by fitting a linear
equation to observed data
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Math TV.com. Facebook, n.d. Web. 15 May
2010. <http://www.mathtv.com/>.
Quick Math. Web Mathmatica, n.d. Web. 15
May 2010. <http://www.quickmath.com/>.