Download a = b

Dineo Black
Addition Property (of
Equality)
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.
http://www.articlesbase.com/k-12-education-articles/addition-property-of-equality-and-inequality-415165.html
Multiplication Property
(of Equality)
For all real numbers a and b , and for c ≠ 0 ,
a = b is equivalent to ac = bc .
http://www.onemathematicalcat.org/algebra_book/online_problems/mult_prop_eq.htm
Reflexive Property (of
Equality)
The property that a = a.
Symmetric Property (of Equality)
The following property: If if a = b then b = a.
http://www.mathwords.com/s/symmetric_property.htm
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
Associative Property of
Addition
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
http://math.about.com/od/prealgebra/a/associative.htm
When three or more numbers are multiplied, the product is the same regardless of the
grouping of the factors. For example (2 * 3) * 4 = 2 * (3 * 4)
http://www.aaamath.com/pro74b-propertiesmult.html
Commutative Property of
Addition
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.
http://www.icoachmath.com/SiteMap/CommutativePropertyofAddition.html
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.)
http://www.multiplication.com/basics/teach2.htm
Distributive Property (of
Multiplication over Addition)
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
http://www.math.unt.edu/mathlab/emathlab/distributive_property_of_multipl.htm
Prop of Opposites or Inverse Property of
Addition
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."
http://wiki.answers.com/Q/Define_property_of_reciprocals
Identity Property of Addition
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.
http://www.northstarmath.com/Sitemap/IdentityPropertiesofAdditionandMultiplication.html
Multiplicative Property of
Zero
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
http://webcache.googleusercontent.com/search?q=cache:uqTe7qZRvcJ:www.kwiznet.com/p/takeQuiz.php%3FChapterID%3D2136%26CurriculumID%3D22+Closure+Property+of+Multiplication&cd=2&hl=en&ct=clnk&gl=us
Product of Powers Property
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.
http://hotmath.com/hotmath_help/topics/product-of-powers-property.html
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.
http://hotmath.com/hotmath_help/topics/power-of-a-product-property.html
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)
http://hotmath.com/hotmath_help/topics/power-of-a-power-property.html
Quotient of Powers
Property
This property states that to divide powers having the same
base, subtract the exponents.
Power of a Quotient
This property states that the power of a quotient can be
Property
obtained by finding the powers of numerator and
denominator and dividing them.
http://www.icoachmath.com/sitemap/Power_Properties.html
Zero Power Property
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
Zero Product Property
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.
http://hotmath.com/hotmath_help/topics/zero-product-property.html
Product of Roots Property
The product of the square roots is the square root of the product.
http://cnx.org/content/m21975/latest/
Quotient of Roots
Property
For any non-negative (positive or 0) real number a and any
positive real number b:
=√a
-√b
http://everyonehatesmath.com/quotient-property-of-square-roots/
Root of a Power Property
Power of a Root
Property
Quiz
Which property?
X+Y=Y+X
Click for answer
Commutative Property (of Addition)
Quiz
Which property?
(5x+9x)+3x=(5x+3x)+9X
Click for answer
Associative Property of Addition
Solving 1st power inequalities
in one variable.
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.
Similarly, we can show that all numbers greater than 4 satisfy this inequality.
http://www.mathsteacher.com.au/year10/ch02_linear_equations/07_subtract/solve.htm
Linear equations in two
variables
Want to learn all about the topic? Well if you click here, you can
learn all you need to know.
Here’s a sample problem: can you graph this: y=x-5?
Linear Systems
Click here to learn about solving linear systems, using the substitution method.
Addition/Subtraction Method
Elimination Method
Can you solve this? 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)
Factoring
Click here to learn about the methods of factoring. Choose
group 1,2,3,4,5, or 6, to the according method you want to
learn.
Can you solve this one: 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)
Rational Expressions
Example
Functions
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.
Click here to learn about functions regarding domain and range.
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.
Click here to learn about graphing parabolas.
Parabolas continued
See if you can graph this one: x2-6x+5
The x-intercepts are (5,0) and (1,0).
y-intercept:
Vertex:
The y-intercept is (0,5).
and
So the vertex is (3, -4).
Now just graph it.
Simplifying Expressions With
Exponents
To learn about this topic with many examples, click here, here, and/or
here
Simplify this:
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.
Your answer is:
http://www.purplemath.com/modules/simpexpo2.htm
Simplifying expressions
with radicals
Click here to learn about this topic. (There are many sample
problems for you to try.)
Try this one:
Word problems
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:
http://www.purplemath.com/modules/mixture.htm
More Word Problems
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.
http://www.purplemath.com/modules/coinprob.htm
More Word Problems
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 solve:
1p + 5p + 10p = 144
16p = 144
p=9
There are nine of each type of coin in the wallet.
http://www.purplemath.com/modules/coinprob.htm
Word Problems
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.
http://www.purplemath.com/modules/ageprobs.htm
Line Best Fit