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1. Definition
2. Example
3. Levels of usage
A. Review of Basic Math and Operations
Variable- a letter that represents a value in an algebraic equation. Ex. x+4=6, in this equation x is the variable.
Coefficient- A number that is multiplies the variable. Ex: 5x+3=12, in this equation 5 is the coefficient.
Constant- A number that stands alone in a problem. Ex: 7x+3; 3 is the constnat.
Absolute Value - the distance from a number to zero.Ex: |-4|=4, in this equation the absolute value of -4 is 4.
Natural Numbers- How we count/(No decimal/fractions/zero) Ex: (1,2,3,4,5...etc)
Whole numbers- natural numbers with zero (no decimals/fractions) Ex: (0,1,2,3,4...etc)
Integers- Whole numbers with negatives (no decimals/fractions) Ex:(...-3,-2,-1,0,1,2,3...)
Rational Numbers - " Predictable" numbers that can be written as a fraction(this includes non-terminating, repeating decimals), ways to express "part" of something. Ex:(1/2 , -45, 9/5, .3333333333, etc) no division by zero.
Irrational Numbers - A decimal that does not end and is not predictable (aka non-terminating, non-repeating)
Ex: ( √2, √11, Pi)
Real Numbers - any number that you can think of that is not in the imaginary category.
Ex: ( Natural, Whole, Integars, Rational, Irrational)
Closure Property - If you take two real numbers and multiply/add them together, then you get another real number.
ex: basic: 2+6=8 advanced: 621(78)= 48,438
Commutative Property - This means that numbers can be added or multiplied in any order
ex: basic: 1+2 = 2+1 advanced: 74(x)(y)(b) = (x)(b)(y)74
Associative Property - If you can group numbers in any way and still recieve the same answer.
ex: basic: 2 x (3 x 4) = (2 x 3) x 4 advanced: (743 x 3) x 756 = 743 x (3 x 756)
Distributive Property - To multiply the term in front of the paranthesis by each of the terms inside the paranthesis
ex: basic: 2(6+3)= 2(90 = 18 advanced: 2( 3x+6) =2(3x) + 2(6) = 6x + 12
Identity Property - Addition - When you add a zero to any number, the sum is that number.
ex: basic: 1+0=1 advanced: 147/3(483)+0=2,367
Multiplication - When you multiply any number by 1, the product is that number.
ex: basic: 1x= x advanced: 754(21)+4(1)= 15,838
Inverse Property - The inverse operation "undoes" the operation.
ex: basic: 1+(-1)=0 advanced: 789+(-263*3) = 0
Reflexive Property - anything is equal to itself (a=a)
ex: basic: 1=1 advanced: 987=987
Symmetric Property - For every number a and b ; if a= b, then b=b
ex: basic: 1=2 then 2=1 advanced: 3x + 5 = 8 and 8= 3x + 5
Property of Equality - Both sides of the equal sign must be the same
ex: basic: 1+1 = 1+1 advanced: 789*4=789*4
Zero Propery of Multiplication - The product of any number and zero is zero
ex: basic:0(5) = 0 advanced: 745(18)+73-157(0)+78
Term - Any combination of numbers/ variables that multiply together.
exs: 3, -4y, -21abc
*They are separated by + (plus) or - (minus) signs
EX: 4x+3y+7y, 8x-3x+4y-2x
Degree - (also know as Order) polynomial with the highest power
B. Solving Equations
To Solve an Equation:
*Find out what perticular variable is equivalent to.
Example: 4x-11=4-x
(Use Math Porperties to change around the equation so that all numbers will be on one side of the equal sign &variables will be on the other )
left right
Your common answer: Variable = Anything and any combination of (numbers, variables, and variable exxpressions)
Example: 4x-11=4-x
(+11) (+11)
4x=15-x
(+1) (+1)
5x=15
5 5
Answer: x= 3 Check: 4(3)-11=4-(3)
12-11=4-3
1=1 *CORRECT!
C. DRT
Distance: The formula for distance is Rate multiplied by time. Distance is meters, miles, inches, etc. An example of a distance problem would be It tooke me one hour and fifeen minutes to drive from here to Ocean City at a speed of 55mph. How far away is Ocean City? To solve this I woud multiply 1.25 (because 15 minutes is a quarter of an hour) by 55 the answer would be 68.8 miles.
Rate: two unrelated types of measurements.
Ex - Miles per Hour; Price per Ounce
Formula for finding the DISTANCE: D=R x T (distance= rate multiplied by time)
Formula for finding the RATE: R=D/T (rate= distance divded by time)
Formula for finding the TIME: T= D/R (time= distance divided by rate)
D. Proving Algebraic Equations
If you have an equation that reads x-2=4 you need facts, rules and statements to be able to prove the equation,
If you were to say something was fun that would be an opinion.
If you were to say the year was 2010 that would be a fact.
To prove your equations you need to know all of your properties.
Closure Property- the sum or product of any two real numbers equaling another real number
Communative Property- numbers and or variables that can be multiplied or added in any sequence
Associative Property- the numbers and or variables that can be grouped together in any way as long as the order is the same.
Distributive Property- the product of a number and a sum is equal to the addition or multiplication of them individualy.
Identity Property of Addition- the sum of any number or variable and zero equals that number or variable.
Identity Property of Multiplication- the product of any number or variable that equals that number of variable.
Inverse Property-to when a number is joined together with its opposite(inverse) to get its identity
Reflexive Property-any real number will equal itself
Symmetric Property- all real numbers are able to be put in any order to equal themselves.
Transitive Property- two numbers that equal the same number are equal to each other.
Multiplication Property of -1- when any whole number is multiplied by negative one, the product is the opposite of itself
Zero Property of Multiplication- the product of any real number or variable and zero will always equal zero.
Property of Equality- if the same one number is added/multiplied/subtracted/divided to both sides of the equation, both sides will remain equivalent.
An equation can go to having seven steps to fifteen steps.
Your first step would be your given equation,
x-2=4 1. Given
You then have to use the inverse property to get the x by itself to solve.
[x-2]+2=[4]+2 2. Inverse Property of Addition
Now you can't forget to label that you are doing the same thing to each side of the equation.
[x-2]+2=[4+2] 3. Property of equality.
Then you regroup your equation to make solving for x easier, but keep it in the same order that you started out with.
x [-2+2]=4+2 4. Associative of Addition
Now you have to preform the action that you are working with, in this case addition.
x [-2+2]=4+2 (-2+2)= 0 and 4+2=6 5. Definition of Addition.
Now you have x by itself on the one side of the equation. And now you have to make sure that you write that since x+0 equals itself.
x+0=6 6. Identity Property of Addition
Now you have your answer to your equation.
x=6 7. Answer
That is an example of a seven step equation.
An example of an equation that would have more steps...
Again your equation is your given
3x+5=7 1. Given
Now again you have to take the first step to get the x by itself.
[3x+5] -5=[7+5] 2. Inverse property of addition
Don't forget what you do to one side you must do to the other.
[3x+5] -5=[7+5] 3. Property of Equality
Now regroup to get rid of the 5.
3x+[5-5]=7-5 4. Associative Property
Now you have to get rid of your 5 by using its inverse.
3x+[0]=2 5. Definition of Addition
Now since you have 3x+0 you know that according to the identity property that you can simplify it.
3x=2 6. Identity Property of Addition.
But you still do not have the x by itself so you have to get rid of the three now by using its inverse like you did with the five.
1/3*(3x)=(2)*1/3 7. Inverse property of multiplication.
Since you multiplied 1/3, the inverse of 3, to one side of an equation you have to do it to the other.
1/3*(3x)=(2)*1/3 8. Property of Equality
Now regroup again to get the inverse and three together.
[1/3*3] +x=2*1/3 9. Associative of Multiplication
Now use multplication to get ride of the three.
(1/3 * 3)=1 and (2* 1/3) = 2/3 1x=2/3 10. Definition of Multiplication
Now according to the identity property you know that anything multiplied by one is itself, so you can simplify this equation.
1x=2/3> x=2/3 11. identity property of multiplication
Now you have your answer to the value of x.
x=2/3 12. Answer.
E. Graphing
slope- describes the steepness of the slope. The higher the slope the steeper the incline. In some equations is represented by the letter m. An example of an equation is y=mx+b.
y-intercept- The point where a line crosses the y axis. If x=0 in a pair of coordinates then the y coordinate is where the y-intercept falls. For example: (0,5) is a y-intercept.
x-intercept- The point where a line crosses the x axis. If y=0 in a pair of coordinates then the x corrdinate is where the x-intercept falls. For example: (5,0) is an x intercept.
negative slope- when a line starts at the top left and falls to the bottom right the slope is negative.
positive slope- when a line starts at the top right and falls to the bottom left the slope is a positive.
undefined slope- when there is no slope it is called undefined. The line is always represented as a vertical line.
System of equations- when you have 2 or more equations with the same variables in them.
x+3y=10 ; 4x-y=14
unique solution- when 2 points intersect on a line the coordinates they intersect is called the unique solution. Ex. if a line intersect at (4,2); (4,2) would be called the unique solution.
Standard form- an equation to find the x and y intercept to graph a line. It is written in a format of Ax+By=C. If the equation was 2x+3y=12 the x-intercept would be (6,0) and the y-intercept would be (0,4)
Parrallel lines- When two lines have the same slope and different intercepts the line is parallel. The equations y=2+4 and y=2+7 are parrallel. The slope of 2 is the same in both equations and the y intercepts are different 4 and 7.
Perpendicular lines- when two lines have the opposite AND reciprocal slope (both conditions). For example, y=4x+8 and y=-1/4x+12 are perpendicular lines because the slope in each are 4 and -1/4, which are opposites and reciprocals of each other.
Line- a straight, never-ending collection of points
Point-Slope Formula- the formula y-y1=m(x-x1) where m=the slope, y1= the given y-coordinate, and x1= the given x-coordinate. When solved, the point-slope formula will give you and equation for a line.
Ex. Given: (3,1) and (3,-7)
Slope: -7-1 = -8 = 0
3-3 = 0
y-1=0(x-3)
y-1=0x-0
y=0x+1
Slope Intercept form: This is our most efficient machine. y=mx+b
H chart: The H chart is our fail proof machine. It gives us lots of points. When using an H chart, you can use any number to plug in as x, but the most common are 2, 1, 0, -1, -2. They are consecutive so they help you make a straight line
Quadrants: Quandrants are the four sections around the graph.
Quadrant 1 is the top right of the graph. The ordered pairs will be (+x, +y) Example- (4,5)
Quadrant 2 is the top left of the graph. The ordered pairs will be (-x, +y) Example- (-3,2)
Quadrant 3 is the bottom left of the graph. The ordered pairs will be (-x, -y) Example- (-2,-1)
Quadrant 4 is the bottom right of the graph. The ordered pairs will be (+x, -y) Example- (6, -3)
Rene Descartes: Created the Cartesian Coordinate system.
Line: A line is a straight, never ending, collection of points. It has an arrow on both ends to show that it keeps going.
Undefined slope: A slope is undefined when it is a vertical line. This happens when an equation doesn't give us a point for y. This means y can be anything. Ex: X = 4 (this means that in every ordered pair x will equal 4. Y can equal anything but we will get a vertical line.)
Zero Slope: The slope is equal to zero when the line is horizontal. This happens when the equation doesn't give us a point for x. This means x can equal anything in the ordered pair. Ex: y= 6 (this means that is every ordered pair y will equal 6. x can equal anything and we will get a horizontal line.)
Delta: Delta means change in. The symbol for a delta is a triangle.
Ordered pair: 2 numbers written in a certain order usuall in parenthesis (9,3) they can show the position on a graph. The x value is first and the y value is second.
corner point- where each line intersects in a shaded region.
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Unique Solution: the point where two lines intersect
Ways of finding unique solutions:
Graph and Check- if they intersect, it is a unique solution.
Say you have the coordinates (4,2) and the equations x + 3y = 10 and 4x - y = 14. Take the x in the coordinate and substitute it into the x spot in the equation, making the equation into 4 + 3y = 10. Then take the y in the coordinate and substitute it into the y spot in the equation, making the equation 4 + 3(2) = 10. After that, multiply 3 and 2 to make 6. Add 4 and 6 to make. In the end 10=10 making the equation a unique solution.
We use this method in graphing.
Substitution- You will start out with 2 equations. Identify which variables (x or y) have a coefficient of 1, and solve for that variable.
example:
4x + 5y = 3
x + 5y = -18 --> x = -5y - 18
4( -5y - 18) + 5y = 3
-20y - 72 + 5y = 3
+72 +72
-15y = 75
-15 -15
y = -5
x = -5(-5) - 18
x = 25 - 18
x = 7
(7, -5)
Linear Combination- eliminating a variable.
example:
9x + 4y = 2
(3x + 4y = -10) -1
9x + 4y = 2
-3x - 4y = 10
6x + 0 = 12
6 6
x = 2
9(2) + 4y = 2
18 + 4y = 2
-18 -18
0 + 4y = -16
4 4
y = -4
(2, -4)
This is used in graphing.
Monomial - Only one term
exs: 1234abcd , 74x , -4
Binomial - Two terms
exs: -22 + 56x, 76-1, 18abcdefghijk+q
Trinomial - Three Terms
exs: 189xyz+9abc-89y, 1+2-3, 6xy+4a-9
Polynomial - Many terms (can be binomials, trinomials, and many terms)
exs: 2xy+67, 483a-64d+2xyz, 1a+2b+3c+4d+5e+6f+7g+8h
Distributive Property of Division for Binomials into Polynomials: Distriubute the binomial into the polynomial by diving the coefficients and exponents of the polynomial by the binomial.
example: 14x^2 + 23x + 3 / 7x +1
14x^2 / 7x = 2x --> 2x * 1 = 2x
14x^2 + 2x -(14x^2 + 2x) = 0 + 21x + 3
21x / 7 = 3 ---> 3 * 1 = 3
21x + 3 - (21x + 3) =0
14x^2 + 23x +3 / 7x +1 = 2x + 3
ANSWER: 2x + 3
A prime number is an integer that has only two factors (1 and itself) and is larger than 1 ex 13
A composite number is an integer with 4 or more factors and must be larger than 1 ex 9
A multiple is a factor that multiplies to give you a product
A factor is multiplied in an addition sentence ex 3 and 5 are factors of 15 (5x3=15)
When using "Synthetic Division" to divide a polynomial by a binomial, the divisor must be:
1. raised to the first power
2. the coefficient must be 1
EX- (4x^3 + x + 7) / (x - 2) NOT- (4x^3 + x + 7) / (2x - 2^3)
Factor By Grouping - A factoring Method that does two sets of Greatest Common Factor(GCF) Factoring.
ex: ux + 8x + uy +8y
Step 1 - Section off 2 terms
(ux + 8x) + uy +8y
Step 2 - Factor out the GCF of those terms
x(u+8) y(u+8)
Step 3- Factor out the shared binomial leaving both GCF's as the terms in your other binomial
(x+y) (u+8)
Composite number- a number greater than one that has at least two or more ways to multiply to get that product.
ex. 12, its multiples are 1, 2, 3, 4, 6, 12.
Prime Factorization- finding all factors of a number that end up prime.
ex. 9
/ \
3 3
GCF- the Greatest Common Factor. When you have two numbers the GCF is the greatest number that goes into both of the numbers.
ex. the GCF of 12 and 6 is 6 since 6 is the highest number that goes into both of the numbers.
order of a polynomial- when the polynomial goes from highest to lowest.
ex. x^2+x+12
not 12+x^2+x
Restrictions on a variable- you have a restriction when there is a variable in the denomenator. There is a restricition in the fraction 4/x.
Undefined fraction- when there is a zero in the Denominator.
Adding Fractions- to do this you need a common denominator, and then you add the numerators together.
Subtracting Fractions- find a common denominator, then subtract the numerators.
Reducing- try to find a factor of both the numerator and denominator that guess in evenly. ex. 15/24; divide the top and bottom by 3 and get 5/8.
Multiply Across- Numerators x Numerators, Denominators x Denominators ex. 5/4 x 3/6
5 x 3=15 ; 4x6=24
15/24
Square Root - The opposite of squaring(raising to a power)
ex: The square root of 16 is 4, The square root of 169 is 13
Adding and Subtracting Fractions - You need to first find a Least common Denominator for the denominator, and you need to change the numerator as well. Then, you add the numerators for addition or subtract the numerators for subtraction, and put this number over the LCD
ex: 3 5 8
- + - = - = 2
4 4 4
Multiplying Fractions - Multiply Across
Numerator Numerator
_________ x _________
Denominator Denominator
ex: 9 7 63
- x - = -
4 13 52
Dividing Fractions - Keep Change Flip (KCF) - To multiply by the reciprocal
ex: 16n 8n 16n 6 4 6 24
__ ÷ __ > __ x __ > __ x _ > _ =
17 6 17 8n 17 2 34
12
__
17
Solving Quadratic Equations
by Gabrielle Konyak
Zero Product Property- Anything multiplied by zero is zero.
Factored Form- having your expression factored
a*b=0 (either a and or b is zero)
First take each factor and set it to zero. Then solve for your variable.
STEPS-
First Step- Make sure your equation is set to equal zero. If not make it by subtracting the number/variable from both sides.
Second Step- Is your equation in factored form? If not then factor it in order to complete the next step.
Third Step- Set each of your factors to zero, then solve for the variable.
x(x-7)=0
First step- Does it equal zero? YES
Second Step- Is it in factored form? YES
Third Step- Set each factor to zero and solve for the vaiable
x=0 x-7=0
+7 +7
x=0 and 7
x^2-2x-3=0
First Step- yes it equals zero
Second step- no, so factor it
(x-3) (x+1) =0
Third Step-
x-3=0 x+1=0
+3 +3 -1 -1
So... x= 3 and -1
CHECK IT!
(3) - 3 (3) +1 = 0
3-3=0 3+1=4
0 multipied by 4 is zero
(-1) -3 (-1) +1=0
-1 -3= -4 -1+1=0
-4 multiplied by zero is zero.
So your answer is correct.
By Alyssa Le
Undefined fraction---> "BOOM" If zero (0) is in the denominator, it will make our expression "undefined."
A square root is the inverse of squaring (raising to the second power)
Ex: The square root of 16= 4; 4²= 16
√ - means what times itself gives you the number under the square root symbol
Ex: √ 49 = 7 ; 7x7=49
Check: Square your answer and that should be the number underneath the √ symbol.
Ex: 7²=49
Exact Square Roots-----> √ 2 ≈ 1.41 ----->Approximate Square Roots( rounded)
Simplify Fractions
√8 (1) is this a perfect square----> No
(2) is there a factor that is a perfect squar----> Yes
√4 x 2 (3) factor our number so we have a P.S. x what't left
√4 x √2 (4) split the radical
2√2 (5) take the square root of the perfect square
2√2 (6) Answer
Raising a number to the ^.5 is the same as taking the square root
Example: √4 = 4^.5
For numbers that are not perfect squares, the square root can be approximated. The method is:
Example: √2
Find any perfect squares surrounding the number
√1= 1
√2
√3
√4= 2
√5
√6
√7
√8
√9= 3
2is between 2 perfect squares, 1 and 4.
This means that the square root of 2 must be between 1 and 2
Next, find the decimal places by examining numbers between 1 and 2
Find the number that, when squared, is less than or equal to 2
1.12 = 1.21
1.22=1.44
1.32=1.69
1.42=1.96
1.52=2.25
The highest result without reaching 3 is 1.96 (1.42)
Repeat the process until 6 decimal places
To simplify rdical:
EX: √108 ask yourself if it is a perfect square. If it is, simply find the (perfect square factor is 36) square root. If not, does it have any factors that are perfect squaes?
Now, factor the number so there is the square root of the perfect square factor (36) times the other factor (3)
6*√3
You can only multipy things that are both under the radical. For example:
(√6*5) [the parenthesis mean the radical continues over it]
√30 = (√6*5)
You can not:
√6 * 7 = √42 it does not equal the square root of 42. It is in simplest form
There are rules you must follow when adding, subtracting, multiplying and dividing fractions.
Adding:
-need a common denominator (but don't forget to change numerator!)
-add the numerators ONLY
Subtracting:
-same as adding fractions but we subtract the numerators
Multiplying:
-multiply across (n*n and d*d)
-look to cross-reduce
Dividing:
-keep the first fraction, change the dividing sign to a multiplication sign, and flip the second fraction so you are using it's reciprocal (KCF)
When you are reducing a fraction, try to find a factor of both the numerator and denominator that goes in each evenly.