Transcript Addition Property (of Equality)

Thinking Algebraically
Nikolas Catalano
Algebra 1
In the upcoming slides, reviewed will be the following
properties;
-
Addition Property (of Equality)
Multiplication Property (of Equality)
Reflexive Property (of Equality)
Symmetric Property (of Equality)
Transitive Property (of Equality)
Associative Property of Addition
Associative Property of Multiplication
Communitive Property of Addition
Communitive Property of Multiplication
Distributive Property
Inverse Property of Addition
Inverse Property of Multiplication
Identity Property of Addition
Identity Property of Multiplication
-
Multiplicative Property of Zero
Closure Property of Addition
Closure Property of Multiplication
Product of Powers Property
Power of a Product Property
Quotient of Powers Property
Power of a Quotient Property
Zero Power Property
Negative Power Property
Zero Product Property
Product of Roots Property
Quotient of Roots Property
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.
Example;
If 3-x=7x, then 3-x+x=7x+x
Adding an “x” to both sides will show the relationship between the two
problems. Add the opposite of the original “x.” The original “x” was a
negative number, so you need to add “x.”
Multiplication Property (of Equality)
In problems with this property, you must basically find the
value of “x.” In doing so, you will have to divide the term
with x into the answer
Example;
5x=3
Divide 5 into the 3, or you can multiply 3 by 1/5 and the
result will be the same.
If 5x=3, then 1/5 x 5x= 1/5 x 3
Reflexive Property (of Equality)
Basically, a number times something is equal to that
number times that something. In ways this is an
“equivalence” property because what is being multiplied
is always what it is simplified in the answer. In cases with
variables, the answer is identical to what is being
multiplied
Example;
3m=3m
Symmetric Property (of Equality)
If and when one term is equal to another term, then the
other term is equal to the first term. In other words, if
a=b, then b=a.
Example;
If a2=b5, then b5=a2
The terms may be in different order, but that does not have
an effect on their equality
Transitive Property (of Equality)
In some instances with 3 terms, they can all “translate” to
one another. Meaning that if the first term equals the
second term and the second term equals the first term,
then the first term will equal the third term
Here is an example of the pairing;
If a2=b5 and b5=c3, the a2=c3
The expression has all equal terms, so you can then pair
off the terms by two, and each of those would be equal
as well
Associative property of Addition
You will find an equation within parenthesis and outside the
parenthesis will be an addition problem to the inner
parenthetic term. In other words, because the order in
which addition goes does not have an effect on the
outcome, you can switch around numbers and find the
same result
Example:
(7+¼)+ ¾ = 7+(¼ + ¾)
As you see, you can rearrange the numbers and end up
with the same answer because of the fact that it is an
addition problem
Associative Property of
Multiplication
The property which states that for all real
numbers a, b, and c, their product is
always the same, regardless of their
grouping: (a . b) . c = a . (b . c)
Example;
(5 x 6) x 7 = 5 x (6 x 7)
Commutative Property of Addition
Here, you can again rearrange the problem,
because it is addition, and find the same
result. You are “commuting” numbers
around, without any effect to the answer
Example;
¼+7+¾=¼+¾+7
The arrangement has no effect on the
answer
Commutative Property of
Multiplication
As you know, if you multiply two numbers, the answer will
be the same no matter what order you multiply the
numbers in. That goes for multiplying with exponents
too.
Example;
a2 x b3 = b3 x a2
The product will always be the same, regardless of the
order of the terms
Distributive Property
When using this property, there will always be a
number before a parenthesis with numbers or
variables inside
4(a+b)
*Note, as you distribute 4 to “a” and 3 to “b”, keep
in mind the positive sign will remain
4a+4b
Inverse Property of Addition
Even with exponents, subtracting a term by
its identical term affixed with the same
exponent will equal zero
Example;
a3+-a3 = 0
In this case, a3 minus a3 will result in nothing
remaining
Inverse Property of Multiplication
If you have a fraction and then multiply the
reciprocal of that fraction, the answer will
always be one
Example;
3/a x a/3 = 1
Why is this? Because your result when
multiplying will be 3a as the numerator and
the denominator
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.
Example;
4+0=4
-10 + 0 = -10
The number keeps its “identity”
Identity Property of Multiplication
A basic rule; multiplying something by one will always result
in that first term. It will keep its “identity” in tact.
a2 x 1 = a2
Even with an exponent, the term will always remain if
multiplied by one
Multiplicative Property of Zero
This property is very basic. Multiplying
something by zero results in zero as the
answer.
Examples;
8x0=0
7x0=0
Closure Property of Addition
This property 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. This property
involves whole numbers.
Example;
a+b=c and b+a=c
5+2=7 and 7+2=9
Closure Property of Multiplication
This property deals mostly with real
numbers. There is not an equation or
expression to represent this property, but if
a is real and b is real, then ab would be a
real number
If a=r and b=r, then ab=r
*r= real number
Power of a Product Property
Within parenthesis are 2 or more numbers or variables.
The exponent outside the parenthesis then applies to
BOTH number or variable.
Example;
(ab)3
Here, you see the exponent, 3, precedes “ab.” With the rule
mentioned above, apply the exponent to “a” and “b”.
a 3b 3
You see the exponent applied to both terms, and this is the
final answer
Power of a Power Property
When multiplying, multiply the powers, then simplify the
base
(a5)3
In other words;
(a)5x3
Simplifying 5x3 will result in
(a)15
Product of Powers Property
When multiplying, add the exponents while retaining the
base number.
Example;
72 × 76
If you know 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
Quotient of a Powers Property
When dividing with exponents, always subtract the
numerator’s exponent by the denominator’s exponent
What you're really doing here is canceling common factors from the numerator
and denominator. Example:
Power of a Quotient Property
Whenever a fraction is in parenthesis and outside the parenthesis lies
an exponent, the exponent should then apply to both the numerator
and denominator
(
)2
Then later applying the 2 to the numerator and denominator will result
as;
a2
b2
Zero Power Property
Any non-zero base to the zero power always
equals 1.
Since x0 is 1 for all numbers x other than 0, it
would be logical to define that 00 = 1.
(4ab)0 =1
Negative Power Property
When the exponent is negative, simply turn it into a fraction by adding
“1” as the numerator. Then, the exponent will become its absolute
value. If necessary, simplify the denominator.
You can, however, easily skip the second step in the equation above.
And get the final result by placing a one as the numerator while
turning the exponent positive.
Zero Product Property
As you know, if a multiplication problem equals zero, one of
the terms must equal zero. If you multiply something by
zero, the answer is zero
Example;
If a2(b-1)=0, then either a2=0 or (b-1)=0
If one of these terms is zero, then the product will be zero
since you are multiplying something by zero
Product of Roots Property
Here, you have two non perfect squares. Since they cannot be
broken down, you must multiply both the 6 and the 15 remaining
under the square root sign
Example;
6
x
15
Find the prime factorization of both 6 and 15. This would leave 3 · 2 · 3 · 5
still remaining under the square root symbol. Cleaning this up would look
like 32 x 10 under the square root symbol
3 10
Quotient of Roots Property
For all positive real numbers a and b, b ≠ 0:
The square root of the quotient is the same
as the quotient of the square roots. The
root applies to both terms
Graphing
Finding solutions for linear equations;
Example: Is (-3,-7) a solution point for f(x) = 2x-1? The
answer is yes because you plug in -3 for “x” and -7 for
f(x).
*F(x) is another word for the “y” value.
Finding slope;
The equation is
Plug the values into each and that is the slope
Graphing cont.
Use the point-slope formula to graph
You can turn it into slope-intercept form by putting the numbers and
variables to the applicable sides.
Factoring GCF
We will look at the following problem and simply find what
each 3 terms have in common in greatest terms.
5x3 – 10x2 – 5x
Including the variables, they each have a 5 and an x in
common. So, you will put this in distributive form (see
slide 12).
5x(x2 – 2x – 1)
As you see, we took out the necessary components of the
original term and put it into the distributive property
Trinomials
- Perfect Square Trinomials
- Reverse Foil
Perfect Square Trinomials
A PST is when the 1st & 3rd terms are squares and the middle term is
twice the product of their square roots.
9x2 – 30x + 25
The 1st and 3rd terms are perfect squares, so inside parenthesis goes
the square roots of both and outside goes the 2nd power
(3x – 5) 2
And this is most definitely a PST because half of the product of the
perfect square’s roots times 2 is the middle term
Reverse FOIL
There will be two parenthesis containing the base products of the inner
and outer terms, The middle term is the sum of the outer and inner
terms multiplied
6x2-17x+12
Here, 3x2 is the 1st term and 4x3 is the 3rd term. Don’t forget to include
the x with the 1st term.
(3x-4)(2x-3)
You can check by redoing the foil process.
Binomials
- Difference of Squares
- Sum or Difference of Cubes
Difference of Squares
With this factoring style, you will again use the GCF method, but you will also
be breaking it down a little further. First, we will find the greatest common
factors for both terms;
75x4 – 108y2
Finding the GCF will result in this
3(25x4 – 36y2)
By using the reverse of foil method by multiplying the outside and inside terms,
you will end up with this conjugate
3(5x2 – 6y) (5x2 + 6y)
Sum/Difference of Cubes
This will deal with exponents to the 3rd power. This is tougher than
squares because you need to get a middle term as well as a 1st and
2nd term.
a3 - b3
Here we see a simple cubed problem with only 2 variables. The base of
the problem is (a-b). The sign remains negative. The (a-b) will go
outside a parenthesis containing the leftovers.
Inequalities
Inequalities compare one number to another with signs. There are 4
signs;
> (greater than)
> (greater than or equal to)
< (less than)
< (less than or equal to)
Example;
7> -7
(7 is greater than negative 7, TRUE)
Graphing Inequalities
Things to remember. The line faces to the left when greater than and to
the right when less than indicating the number is beyond the
comparison point. The circle where you begin on the graph is closed
if the number compared can be equal to and is open if it is greater or
less than in general
Example;
X<3
(x is less than 3)
Parabolas
To graph a parabola, you should know the vertex, x and y intercepts, and the
axis of symmetry). To find the coordinates of the vortex, the equation of the
first coordinate is
b
2a
B is the 2nd term of the given parabolic equation while a is the 1st term. This
equation will also find the line of symmetry. You will find the second part of
the vertex coordinates by plugging in what you find in the given equation.
You will find the x-intercept coordinates by setting y=0. When you find the xintercept you can plug that in to find the y-intercept. The parabola will open
up if a in the equation is positive
Functions
Discrete means you have to “lift your pencil”
to continue graphing. Continuous means
you can draw the graph in one motion.
Range is found by reading the y coordinate
of the graph
Domain is found by reading the x
coordinates of the graph
Quadratic Formula
This formula will help you find the x value
Links for Practice
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http://www.mathtv.com (Recommended)
http://mathworld.wolfram.com/topics/Algebra.html
http://www.coolmath.com/algebra/Algebra1/index.html
http://www.quickmath.com/
http://www.math.com/
http://www.brainpop.com/math/algebra/
http://nlvm.usu.edu/en/nav/topic_t_2.html
http://sun.cs.lsus.edu/webMathematica/rmabry/factortrinomial.jsp
http://www.internet4classrooms.com/gateway_algebra.htm