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GRE Quantitatuve Reasoning > Systems of Equations > Flashcards

Systems of Equations Flashcards

QUANT CALCULATIONS

1
Q

ALGEBRA
Standard Form

A

ax + by = c

a, b, and c = intergers
x - x and y intercepts, respectively.

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2
Q

ALEGEBRA
GRAPHING LINEAR INEQUALITIES
Slope Intercept Form

A

y = mx + b

m = slope

b= y intercept

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3
Q

ALGEBRA
Quadratic Formula

A

x = -b +or- √ bˆ2 - 4ac/ 2(a)

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4
Q

ALGEBRA
Polynomial Rational Expression

A

r-4/r^2 - 5r + 6
Multiplication: Factor, Reduce, Multiply
Division: Factor, Flip, Reduce, Multiply
Addition and Subtraction: Requires LCDs

The procedure for subtracting polynomials, is to multiply the second expression by -1 and add the two expressions together.

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5
Q

ABSOLUTE VALUE
Inequalities

A

Absolute Values are always zero or a positive number

Absolute Value is greater than the compound is an “or” inequality

Absolute values if less than the compound is an “and” inequality

When undoing (rewriting to solve linear equations) an absolute value, you must flip the inequality symbol on the negative solution.

ı7x-1ı > -5 infinite solution
When an absolute value is greater than a negative number, there are infinite solutions.

ı7x-1ı < -5 no solution
When an absolute value is less than a negative number, there are no solutions.

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6
Q

ARITHMETIC
Simple Interest

A

A = P(1 + RT)

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7
Q

ARITHMETIC
Compound Interest

A

A = P[1 + r/n]ˆnt
n = number of periods
nt - periods in given time

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8
Q

PLANE GEOMETRY
Measures in Angels

A

Vertical Angels will always be congruent Parallel Lines required to determine measure among Corresponding and Alternate Interior (congruent therefore: < = <)

and

Exterior, and Consecutive Interior (supplemental therefore: < + < = 180) Angles

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9
Q

PLANE GEOMETRY
Circles and Cylinders
(Circumference, Area, Radius, Volume)

A

CIRCLES
A = π(rˆ2)
C = 2 π R or C= π D
R = C/(2π)
V = (4/3 π r^3/3) * 2 and Hemisphere = (2/3/ * π r^3)/3

HEMISHPERE
V = 2/3π OR r^2/3

CYLINDERS
C = 2πR
R = C/(2π)
A = πR^2
V = h(πr^2)

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10
Q

PLANE GEOMETRY
Triangles and Pyramids
(Area, Volume (Pyramid), Hypotenuse)

A

TRIANGLES
Area = 1/2 Base * Height
Hypotenuse = a^2 + b^2 = c^2

PYRAMID
V = 1/3 (Base^2)(Height)

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11
Q

PLANE GEOMETRY
Rectangles
(Area, Diameter, Perimeter)

A

RECTANGLES
Area = LW

Diagonal = Sq Rt of L^2* * W^2

Perimeter = 2L * 2W

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12
Q

PLANE GEOMETRY
Squares and Cubes
(Area, Perimeter, Diagonal)

A

SQUARES
A = sˆ2
P = s*4
D = √S1 + √S2

CUBES
V = A/#S
V = LWD
V = s^3

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13
Q

PLANE GEOMETRY
Triangle Similarity Theorems and Formulas

A

AA Angle Angle: two angles in the triangle are provided

SAS Side Angle Side: two sides and the included angle of both sides triangles.
Similarity Test:
1. Included angle must be congruent;
2. Corresponding sides must be proportional by decimal.
Area = (1/2) × side1 × side2 × sin (included angle)

SSS all three sides lengths for both triangles
Similarity Test:
1. Corresponding sides must be proportional by decimal.

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14
Q

PLANE GEOMETRY
Pythagorean Theorem

A

Applies Only to Whole #s

The sum of the squares of two sides of a right trained is equal to the sum of the square of the hypothenuse: Side opposite right triangle

Aˆ2 + Bˆ2 = Cˆ2

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15
Q

ALGEBRA
Inequalities

A

Greater Than or Less Than

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16
Q

ALGEBRA
Velocity Point Slope Formula

A

VELOCITY POINT SLOPE FORMULA
Y = Y1 + m(X - X1)
Y1 = change in Y
X1 = change in X

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17
Q

ALGEBRA
The Distance Formula

A

DISTANCE FORMULA
c= √aˆ2 - bˆ2
d = √(X2 - X1)ˆ2 + (Y2 - Y1)ˆ2

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18
Q

ALGEBRA
The MidPoint Formula

A

MIDPOINT FORMULA
Midpoint = X1 + X2/2, Y1 + Y2/2

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19
Q

ALGEBRA
Transformation Formulas

A

TRANSFORMATION FORMULAS
y = f(x) + n
x = f(x)

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20
Q

ALGEBRA
Translations add or subtract values from a function

A

f(x - h) + k
h = x axis
k = y axis

f(x) + 1
f(x-2) + 4
rotation: spin
reflection: flip x-line
delation: stretch

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21
Q

ALGEBRA
ABDOLUTE VALUES
Standard Form Inequalities
Standard Form Equation

A

STANDARD FORM EQUATION
y = aıx-hı + k
a= slope
h = x axis
k = y axis

Absolute Value Equations:
always result in two answers;

The most common way to represent the absolute value of a number or expression is to surround it with the absolute value symbol: two vertical straight lines.

|6| = 6 means “the absolute value of 6 is 6.”
|–6| = 6 means “the absolute value of –6 is 6.”
|–2 – x| means “the absolute value of the expression –2 minus x.”
–|x| means “the negative of the absolute value of x.”

https://www.hmhco.com/blog/teaching-absolute-value-of-a-number-in-math

Absolute value signs around a mathematical operation should be treated like parentheses, which means you we should perform the operations inside the symbols first.

The first thing to do while solving the equation is to split the equation into two separate equations.

|x| must be positive

STANDARD FORM INEQUALITIES

“OR” COMPOUNDS
x > n OR x < -n
Arrows on graph go in opposite directions and the circle point is open.

“AND” COMPOUNDS
n lxl n ≥ n
x = ≥ n
Graph line is solid with closed circle points.

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22
Q

DATA ANALYSIS
Measures of Central Tendency

A

Mean, Median, Mode, Range

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23
Q

DATA ANALYSIS
Probability Equation Formula

A

favorable outcomes/# total outcomes

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24
Q

DATA ANALYSIS
Compound Events

A

involves the probability of more than one event happening together

Use Probability Formula and then multiply each of the probabilities together

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25
Q

DATA ANALYSIS
Complementary Events

A

add probabilities together to equal a whole

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26
Q

DATA ANALYSIS
PROBABILITES
Calculating Combinations
(Factorials)

A

where order of outcome does not matter

number of favorable outcomes/number of total outcomes

nCr = n!/r!(n-r)!

n = Toal number of items
r = number of items being chosen at a time
C = combination

factorials = = n!

4! = 4x3x2x1

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27
Q

ALGEBRA
Factorials

A

product of all the positive integers equal to and less than your number

to calculate multiply all positive integers ≤ 4

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28
Q

DATA ANALYSIS
PERMUTATIONS
Where Order Matters
(Factorials)

A

a method to calculate the number of events occurring where order matters

nPr=n!/(n-r)!
then calculate total # for outcomes

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29
Q

DATA ANALYSIS
Probability

A

a ratio that states the likelihood of an event happening.

30
Q

DATA ANALYSIS
Either/Or Probability Formula
(Non-Overlapping Events)

A

P(A or B) = P(A) + P(B)

Favorable Events/Total Events

31
Q

DATA ANALYSIS
Standard Deviation

A

the measure of how closely all the data in a data set surrounds the mean.

32
Q

DATA ANALYSIS
Normal Distribution

A

is represented when the majority of the data is found close to the average of the set.

most of your data will be within one standard deviation of the mean, Therefore, add and subtract the standard deviation to and from the mean.

33
Q

DATA ANALYSIS
PROBABILITY & STATISTICS
Variance and Standard Deviation

A

is the average of each distance from the mean squared

  1. average the data points
  2. subtract mean from data point
  3. square the each of the differences
  4. average the products (Variance)
    5.√variance for standard deviation
34
Q

ALGEBRA
(Graphing) Quadratic Equations

A

ax^2+bx+c=0

35
Q

ALEGEBRA
Exponential Functions Formula
(Rate of Growth)

A

y=ab^x

y = output
a = original value of the function
b = rate of growth (increase in value)
x = the input

36
Q

ARITHMETIC
RATES
Simple Interest (usually loans for one year or less)

A

V= P(1+r/100)

I=PT

I- interest
P-%
T-time

37
Q

ARITHMETIC
RATES
Compound Interest

A

Compounded Several Times/Year:
V = P (1 + r/n)^nt

Compounded Annually:
V=P(1+ (r/100n))^nt

P = principal
r = rate
n= times interest is added
nt = number of periods

38
Q

ARITHMETIC
Rate Formula

A

Anything that can be gained or loss over time

rate = gain or loss/time

OR

gain or time = rate*loss

speed: g/l is distance
earnings: g/l earnings over time

set up table to fill in knowns
solve using algebra and substitutions

39
Q

ALEGEBRA
Slope Equation

A

y2-y1/x2-x1 = Slope

40
Q

ALEGEBRA
Graphing Circles

A

(x-y)^2 + (y-b)^2 = r^2

41
Q

DATA ANALYSIS
PROBABILITY & STATISTICS
Calculating Percent Increases

A

Precent Increase = (Frequency 2 - Frequency 1/ Frequency 1)*100

42
Q

DATA ANALYSIS
PROBABILITY & STATISTICS
At Least Once Rule (Independent Events)

A

The probability of an event occurring + the probability of an event not occurring = 1 or 100%

Use exponents if the event effect more than one data set (item or person)

43
Q

DATA ANALYSIS
PROBABILITY & STATISTICS
Conditional Probabilities (Dependent Events)

A

To get the probability of the final event occurring:

  1. Multiply the probability of each event
  2. Simply the product
44
Q

DATA ANALYSIS
PROBABILITY & STATISTICS
Quartiles

A

Q1 = median of the smallest set to the median of the median
Q2 = median of the whole set
Q3 = median of the largest set to the median of the median

45
Q

PROBABILITY & STATISTICS
Percentile Rank Formula

A

R=P/100(n+1)

R represents the rank order of the score.

P represents the percentile rank.

N represents the number of scores in the distribution..

46
Q

DATA ANALYSIS
Range

A

Range = Highest Value - Lowest Value

47
Q

DATA ANALYSIS
FREQUENCIES
Relative Frequency

A

The number of times a specific event occurs divided by the total number of events that occur

This is a good predictor of an event occurring in the future

Calculating Percent Increases

48
Q

DATA ANALYSIS
FREQUENCIES
Cumulative Frequency

A

The total number of times a specific event occurs within the time frame given

Calculating Percent Increases

49
Q

COORDINATE GEOMETRY
Cartesian Plane

A

XY Plane
Ordered Pair: A set of coordinates shown as two numbers inside parentheses (x,y).
Origin: (0,0)

planation
The point where the lines meet provide the solution to a system of equations.

Parallel Lines: have no System of Equation to solve a problem because there are no intersecting lines.

A system of equations must have more than one equation in a single problem.

50
Q

COORDINATE GEOMETRY
Quadrants

A

Q1 (+,+)
Q2 (-,+)
Q3 (-,-)
Q4 (+,-)

51
Q

COORDINATE GEOMETRY
Slopes and Tangents

A

y=mx+b
if m = (-), then slope = (-) and line is slanted downward
if m = (+), then slope = (+) and line is slanted upward

The larger the number, the steeper the slope

UNDEFINED
Vertical Line Parallel to the Y-Axis (x=-)
Horizontal Line Parallel to the X-Axis (m=0) y=0x+b
b = y intercept
(y=-)

TANGENTS
Slopes with a single point along a curve (no straight line)
y = x^2 (smile)
y = xsin(x) (wavy)
curve = 0

52
Q

COORDINATE GEOMETRY
The Distance Formula

A

A Condensed Form of The Pythagorean Theorem
d = √(x2-x1)^2 + (y2-y1)^2

53
Q

COORDINATE GEOMETRY
The Midpoint Formula

A

Horizontal and Vertical Lines: Count and Divide by 2

Diagonal Lines:
Midpoint = (x1+x2/2, y1+y2/2)

54
Q

COORDINATE GEOMETRY
Transformations and Absolute Values
STANDARD FORM ABSOLUTE VALUE EQUATION

A

STANDARD FORM ABSOLUTE VALUE EQUATION
y = alx-hl+k
a = slope
h = left/right
k = up/dpwn

Absolute Values that equal negative numbers have no solution because absolute values are always positive.

Transformations; How to Shift Graphs on a Plane

Types of Shifts:
Rotations (spin a function); Reflections (flip a function across a line);
Dilations (expand or contract a function);
Translations (slide a function around w/o changing its size)

Translations - accomplished by adding or subtracting from the function. f(x + or - h) + or - k. Adding or subtracting inside f(x) shifts the graph right or left, respectively.
Adding or subtracting outside f(x) shifts the graph left or right, respectively.

Write equations. for g in terms of f: g(x) = f(x)

Second Most Common Transformation: Dilations (expand or contract a function) using lXl making the wider or thinner. y = mlXl
m tells you the new slope in V
y = 2lXl go up two and over one in each direction creating a narrow V
y = (1/3)lXl go up one and over three in each direction creating wide V (numerator = Y, denominator = X)

Reflections (flip a function across a line) using lXl to create a mirror image resulting in a V-line with a vertex at the bottom. ABSOLUTE VALUES turn (=)#s back into (+)#s.
y = -mlXl
m tells you the new slope in V
y = -4lXl go down four and over one in each direction creating a narrow reflective V
y = (-5/2)lXl go down five and over two in each direction creating wide reflective V

55
Q

COORDINATE GEOMETRY
Perpendicular Lines

A

Perpendicular Lines have slopes that are opposite reciprocals.

One slope should be positive, The other should be negative, One of them should be the flipped version of the other.

For Example:
Since the first slope is positive three-halfs, then the second slope should be negative two-thirds.

56
Q

COORDINATE GEOMETRY
Vertex

A

Any line above or below y = 0 would be parallel to the x-axis.

57
Q

COORDINATE GEOMETRY
Point-Slope Formula

A

Change in Time(Speed) = Change in Location

Used to find the line that crosses an ordered pair

y = y1 + m(x - x1)

y1 - change in y
m = slope
(x - x1) = change in x

58
Q

MATHEMATICS
SQUARE ROOTS
√ Rules and Problems

A

Principal √ are the positive √ of numbers.

Perfect Square is when the principal √ is an integer.

SIMPLIFIED √:
1. The radicand has no square factors other than 1 (perfect square);
example √200 = √2 * 100 = 10 * √2
2. There are no √ signs in the denominator.

Combine terms with the same radicand (number beneath the √ sign).

59
Q

MATHEMATICS
Scientific Notation

A

Proper scientific notation has the decimal right after the first non-zero digit followed by a multiplication by 10 to a power.

Move the decimal place from its current location to the space directly behind the first non-zero digit in the number.

60
Q

FUNCTIONS
Range (Outputs) v. Relations

A

Range corresponds to the Y-axis values (dependent variables).

The range of a function includes all of its possible outputs OR all of its possible n values.

A function has one output for each input. If the same input produces more than one output, then it is a relation.

61
Q

FUNCTIONS
Composites

A

A composite of two functions means to take the output from one function (‘inside’ function), which becomes the input for the other function (‘outside’ function).

For Example, if you have f(g(x)), then you assign a value to g, then take that output and put it into f.

62
Q

FUNCTIONS
Inverse and Line Test

A

In order to solve an f(x) problem you must know the value of function of x.

If a function is the inverse of another, then the composite of the two functions will be equal to ‘x’.

The inverse function does the opposite (inverse) of all of the operations done by the original function, but in the opposite order.

EXAMPLE 1, if you have f(x) = 2x - 3,

you need the parenthesis around the (x + 3) as the whole side is divided by 2 or f -1 (x) = (x + 3)/2. You must divide the 3 and the x by the 2.

EXAMPLE 2
where you have f(x) = x/8 + x + 2, then:

  1. Write the following equation:
    y = x/8 + x + 2, :then:
  2. Flip the y and x, which gives you:
    x = y/8 + y + 2
  3. Subtract 2 from both sides:
    x - 2 = (1/8)y + y
  4. Add the y’s on the right side:
    x - 2 = (9/8)y
  5. Multiply each side by 8/9:
    8/9(x - 2) = y to get:

f -1 (x) = (8/9)(x - 2)

EXAMPLE 3:
y = 3(x - 2)^3

Step 1: Switch the x and y to get:
x = 3(y - 2)^3

Step 2: Now solve for y.
Start by dividing each side by 3.
x/3 = (y - 2)^3

Step 3: Take the cube root of both sides:
(x/3)^(1/3) = (y - 2)

Step 4: Now add 2 to both sides:
(x/3)^(1/3) + 2 = y

Therefore, f^-1(x) equals
(x/3)^(1/3) + 2 = y

LINE TEST
Any function can have only one output>
Vertical Line Test (originate function):
If it crosses more than one place, the it is not a function.
If it crosses only one spot at a time then it is a function.

Horizontal Line Test (inverse functions):
If it crosses more than one place, the inverse is not a function.
If it crosses only one spot at a time then it is a function.

63
Q

FUNCTIONS
Independent and Dependent Variables

A

INDEPENDENT
An independent variable is the variable you can manipulate.

DEPENDENT
A dependent variable is the variable that changes as a result of the independent variable manipulation. It’s the outcome you’re interested in measuring, and it “depends” on your independent variable.

VARIABLES IN RESEARCH
Researchers often manipulate or measure independent and dependent variables in studies to test cause-and-effect relationships.

The independent variable is the cause. Its value is independent of other variables in your study.

The dependent variable is the effect. Its value depends on changes in the independent variable.

64
Q

FUNCTIONS
Ordered Pairs, Domains (Inputs)

A

ORDERED PAIRS
An ordered pair is a composition of the x coordinate (abscissa) and the y coordinate (ordinate), having two values written in a fixed order within parentheses.

Every x input has to have a unique y output.

DOMAIN
Domain corresponds to the X-axis values (independent variables).

The domain of a function f(x)
is the set of all values for which the function is defined, and the range of the function is the set of all values that f takes.

Domains cannot have a zero denominator, nor can it have the square root of a negative number.

(In grammar school, you probably called the domain the replacement set and the range the solution set. They may also have been called the input and output of the function.)

For Example,
Consider the relation {(0,7),(0,8),(1,7),(1,8),(1,9),(2,10)}.
Here, the relation is given as a set of ordered pairs. The domain is the set of x -coordinates, {0,1,2}, and the range is the set of y -coordinates, {7,8,9,10}. Note that the domain elements 1 and 2 are associated with more than one range elements, so this is not a function.

65
Q

FUNCTIONS
Composition of Functions

A

The composition of functions is the process of combining two or more functions into a single function.

For Example, if f(x) = 5 - 2x and g(x) = -4x + 2, then
g(f(x)) = -4(5 - 2x) + 2
= -20 + 8x + 2
= 8x - 18
= 2(4x - 9)

66
Q

ALGEBRA
PARABOLAS

A

Vertex form is set up as: y = a(x - h)2 + k

How to Graph a Parabola of the Form Y = a(x-h)^2 + k
Step 1: Identify h and k
from the equation, being careful to pay attention to the sign (positive or negative) on each number. The point
(h, k) is the vertex of the parabola. For example, y=3(x+6) 2−7 can be written as y=3(x−(−6))2+(−7) so that it is clear that (−6,−7) is the vertex.

Step 2: Find an additional point on the graph of the parabola by plugging a value for x other than h into the equation and solving for y. Often, the easiest choice is x=0, which leads to finding the y-intercept, the place where the graph crosses the y-axis.

Step 3: Plot the vertex (h,k) found in Step 1 and the additional point found in Step 2. Then use the facts that the graph of a parabola has a valley or peak at the vertex and is symmetric about the vertical line through the vertex to sketch the graph.

67
Q

ALGEBRA
COMPOUND INEQUALITIES

A

“AND”
When the absolute value is less than, the compound is an “AND” inequality. A compound inequality is where both conditions must be satisfied is called an ‘and’ compound inequality.
1. In order for the solution to work, both conditions must be true.
2. It has a less than symbol.

“OR”
When the absolute value is greater than, the compound is an “OR” inequality. The arrows of the inequality go in opposite directions.
An ‘or’ compound inequality is when the arrows of an inequality go in opposite directions.

68
Q

ALBGEBRA
INTERVAL NOTATIONS

A

PARENTHESES
Interval notations are set up using parentheses. Because the range does not include 3 or 97, the set is bound by parentheses.

BRAKETS

69
Q

ALGEBRA
SOLVING 1-VARIALBLE INEQUALITIES

A

SOLVING 1-VARIALBLE INEQUALITIES

Remember to flip the inequality sign when multiplying or dividing by a negative number.

70
Q

ALGEBRA
GRAPHING 1-VARIALBLE INEQUALITIES

A

Form a straight line plane, with either open or solid circles on the variable.

An open circle indicates that the variable is not included in the solution.

A solid circle indicates that the variable is included in the solution.

71
Q

ALGEBRA
GRAPHING 2-VARIALBLE INEQUALITIES

A

Form a vertex on the Cartesian plane, with either the inside or the outside of the V shaded.

A dotted line indicates that the ordered pairs are not included in the solution.

A solid line indicates an ‘or equal to’ inequality on a graph.

GRE Quantitatuve Reasoning (2 decks)

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