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GRE Math > Part 2: Algebra > Flashcards

Part 2: Algebra Flashcards

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1
Q

What is a variable?

A

A letter that represents an unknown value

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2
Q

What is an algebraic expression?

A

an arithmetic expression with one or more variables

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3
Q

What is a term?

A

It’s a single part of an algebraic expression that is separated by an addition or subtraction sign from other parts of the expression.

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4
Q

What are like terms?

A

Terms are alike when they contain the same variables, with the same exponents.

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5
Q

What is a constant?

A

A term that does not contain any variables.

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6
Q

What is a coefficient?

A

A number that is multiplied by variables.

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7
Q

What is a polynomial?

A

An algebraic expression with a finite number of terms.

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8
Q

What is the degree of a term?

A

It’s the exponent that’s on the variable.

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9
Q

What is the degree of a polynomial?

A

The greatest exponent in the polynomial.

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10
Q

Identity 1: ca + cb

A

c(a + b)

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11
Q

Identity 2: ca - cb

A

c(a - b)

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12
Q

Identity 3: (a + b) ^2

A

a^2 + 2ab + b^2

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13
Q

Identity 4: (a - b)^2

A

a^2 - 2ab + b^2

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14
Q

Identity 5: a^2 - b^2

A

(a + b)(a - b)

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15
Q

Identity 6: (a + b) ^3

A

a^3 + 3a^2b + 3ab^2 + b^3

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16
Q

Identity 7: (a - b)^3

A

a^3 - 3a^2b + 3ab^2 - b^3

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17
Q

Rule 1: x^-a

A

1/x^a

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18
Q

Rule 2: (x^a)(x^b)

A

x^(a+b)

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19
Q

Rule 3a: x^a/x^b

A

x^(a-b)

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20
Q

Rule 3b: x^(a-b)

A

1/x^(b-a)

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21
Q

Rule 4: x^0

A

1

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22
Q

Rule 5: (x^a)(y^a)

A

(xy)^a

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23
Q

Rule 6: (x/y)^a

A

(x^a)/(y^a)

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24
Q

Rule 7: (x^a)^b

A

x^(ab)

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25
Q

What is an equation?

A

It is a statement of equality between two expressions.

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26
Q

What does it mean to solve an equation?

A

It means to find the values of the variables that satisfy the equation.

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27
Q

What are equivalent equations?

A

They are equations that have the same solution.

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28
Q

Equivalent equation rule #1 regarding constants:

A

The same constant can be added or substracted from each side of an equation, equality is preserved and the new equation is equivalent to the old one.

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29
Q

Equivalent equation rule #2 regarding constants:

A

The same constant can be multiplied or divided on each side of an equation and the new equation will be equivalent to the original one.

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30
Q

Equivalent equation rule #3 regarding equivalent expressions:

A

An equivalent expression can replace an expression inside an equation and the new equation will be equivalent to the original one.

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31
Q

What is a linear equation?

A

It’s a 1st degree polynomial where each term is either a constant or a single variable multiplied by a coefficient. None of the variables are multiplied by each other.

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32
Q

What is a linear equation in two variables?

A

ax + by = c

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33
Q

What is an ordered pair?

A

An ordered pair is a solution for a linear equation in two variables.

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34
Q

What is a quadratic equation?

A

ax^2 + bx + c = 0

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35
Q

What is the quadratic formula?

A

The quadratic formula is used to solve a quadratic equation: x = (-b +- √(b^2 - 4ac))/(2a)

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36
Q

What is the formula for simple interest?

A

V = P(1 + rt/100)

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37
Q

What is the formula for compound interest, compounded annually?

A

V = P(1 + r/100)^t

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38
Q

What is the formula for compound interest, compounded n times per year?

A

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

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39
Q

How do you determine the distance between two points on the coordinate system?

A

Use the pythagorean theorem: a^2 + b^2 = c^2, where c is the hypotenuse.

40
Q

In the equation y = mx + b, what are m and b called?

A

m is the slope, b is the y-intercept.

41
Q

What shape is a parabola where a is positive in a quadratic equation?

A

The parabola opens up and its vertex is its lowest point.

42
Q

What shape is a parabola where a is negative in a quadratic equation?

A

The parabola opens downward and its vertex is its highest point.

43
Q

What is the equation that graphs a circle?

A

(x - a)^2 + (y - b)^2 = r^2, where (a, b) is the center point, and r is the radius.

44
Q

What does the graph of the absolute value function h(x) = |x| + C look like?

A

A “V” shape with its vertex at 0. Or if there is a constant C, then the vertex is at y = C.

45
Q

What does the positive square root function graph look like?

A

The upper half of a parabola lying on it side.

46
Q

How do you achieve a reflection about the y=x line?

A

By interchanging x and y.

47
Q

Describe the graph made by the function g(x) = (x + 1)^2

A

It’s a parabola with the vertex at the origin, but then shifted to the left 1 unit.

48
Q

Describe the graph made by the function ch(x), for any constant c, in relation to the function h(x).

A

It’s the graph h(x), stretched vertically if c > 1. And then shrunk vertically, if 0 < c < 1

49
Q

Find an algebraic expression for:

The square of y is subtracted from 5 and the result is multiplied by 37.

A

(5 - y^2) * 37

50
Q

Find an algebraic expression for:

Three times x is squared, and the result is divided by 7

A

((3x)^2) / 7

51
Q

Find an algebraic expression for:

The product of (x + 4) and y is added to 18.

A

y(x + 4) + 18

52
Q

Simplify each of the following algebraic expressions:

3x^2 - 6 + x + 11 - x^2 + 5x

A

2x^2 +6x + 5

53
Q

Simplify each of the following algebraic expressions:

3(5x - 1) - x + 4

A

14x + 1

54
Q

Simplify each of the following algebraic expressions:

x^2 - 16 / x - 4, where x != 4

A

x + 4

55
Q

Simplify each of the following algebraic expressions:

2x + 5)(3x - 1

A

6x^2 + 13x -5

56
Q

What is the value of f(x) = 3x^2 - 7x + 23, when x = -2?

A

49

57
Q

What is the value of h(x) = x^3 - 2x^2 + x - 2, when x = 2?

A

0

58
Q

What is the value of k(x) = (5/3)x - 7, when x = 0?

A

-7

59
Q

If the function g is defined for all nonzero numbers y by g(y) = y/|y| , find the value of each of the following:
g(2)

A

1

60
Q

If the function g is defined for all nonzero numbers y by g(y) = y/|y| , find the value of each of the following:
g(-2)

A

-1

61
Q

If the function g is defined for all nonzero numbers y by g(y) = y/|y| , find the value of each of the following:
g(2) - g(-2)

A

2

62
Q

Simplify: (n^5)(n^-3)

A

n^2

63
Q

Simplify: (s^7)(t^7)

A

(st)^7

64
Q

Simplify: r^12/r^4

A

r^8

65
Q

Simplify: (2a/b)^5

A

32a^5/b^5

66
Q

Simplify: (w^5)^-3

A

w^-15

67
Q

Simplify: (5^0)(d^3)

A

d^3

68
Q

Simplify: ((x^10)(y^-1)) / ((x^-5)(y^5))

A

(x^15)(y^-6)

69
Q

Simplify: (3x/y)^2 / (1/y)^5

A

9x^2y^3

70
Q

Solve for x: 5x -7 = 28

A

x = 7

71
Q

Solve for x: 12 - 5x = x + 30

A

x = -3

72
Q

Solve for x: 5(x + 2) = 1 - 3x

A

x = -9/8

73
Q

Solve for x: (x + 6)(2x - 1) = 0

A

x = -6 or x = 1/2

74
Q

Solve for x: x^2 + 5x - 14 = 0

A

x = 2 or x = -7

75
Q

Solve for x: x^2 - x - 1 = 0

A

x = (1 + √5)2 and x = (1 - √5)/2

76
Q

Solve for x and y:
x + y = 24
x - y = 18

A

x =21, y = 3

77
Q

Solve for x and y:
3x - y = -5
x + 2y = 3

A

x = -1, y = 2

78
Q

Solve for x and y:
15x - 18 - 2y = -3x + y
10x + 7y + 20 = 4x + 2

A

x = 1/2, y = -3

79
Q

Solve for x:

-3x > 7 + x

A

x < -7/4

80
Q

Solve for x:

25x + 16 >= 10 - x

A

x >= -3/13

81
Q

Solve for x:

16 + x > 8x -12

A

x < 4

82
Q

For a given two-digit positive integer, the tens digit is 5 more than the units digit. The sum of the digits is 11. Find the integer.

A

83

83
Q

If the ratio of 2x to 5y is 3 to 4, what is the ratio of x to y ?

A

15: 8

84
Q

Kathleen’s weekly salary was increased by 8 percent to $712.80. What was her weekly salary before the increase?

A

660

85
Q

A theater sells children’s tickets for half the adult ticket price. If 5 adult tickets and 8 children’s tickets cost a total of $81, what is the cost of an adult ticket?

A

9

86
Q

Pat invested a total of $3,000. Part of the money was invested in a money market account that paid 10 percent simple annual interest, and the remainder of the money was invested in a fund that paid 8 percent simple annual interest. If the total interest earned at the end of the first year from these investments was $256, how much did Pat invest at 10 percent and how much at 8 percent?

A

$800 at %10

$2200 at %8

87
Q

Two cars started from the same point and traveled on a straight course in opposite directions for 2 hours, at which time they were 208 miles apart. If one car traveled, on average, 8 miles per hour faster than the other car, what was the average speed of each car for the 2-hour trip?

A

car 1: 48m/h

car 2: 56m/h

88
Q

A group can charter a particular aircraft at a fixed total cost. If 36 people charter the aircraft rather than 40 people, then the cost per person is greater by $12.

(a) What is the fixed total cost to charter the aircraft?
(b) What is the cost per person if 40 people charter the aircraft?

A

Total cost: $4320

CPP for 40: $108

89
Q

An antiques dealer bought c antique chairs for a total of x dollars. The dealer sold each chair for y dollars.

(a) Write an algebraic expression for the profit, P, earned from buying and selling the chairs.
(b) Write an algebraic expression for the profit per chair.

A 
P = cy - x 
PPC = y - x/c
90
Q

Algebra Figure 16 that follows shows right triangle PQR in the xy-plane. Find the following where P is (-2, 6), R (5,0), and Q is unknown but is at y = 0 and x -2.

(a) The coordinates of point Q
(b) The lengths of line segment PQ, line segment QR, and line segment PR
(c) The perimeter of triangle PQR
(d) The area of triangle PQR
(e) The slope, y-intercept, and equation of the line passing through points P and R

A 
Q (-2, 0)
PQ =6 
QR = 7
PR = √85
perimeter = 13 + √85
area = 21
y = -6/7x  + 30/7
91
Q

In the xy-plane, find the following.

(a) The slope and y-intercept of the line with equation 2 y + x = 6
(b) The equation of the line passing through the point (3, 2) with y-intercept 1
(c) The y-intercept of a line with slope 3 that passes through the point (−2, 1)
(d) The x-intercepts of the graphs in parts (a), (b), and (c)

A 
slope = -1/2
y-intercept = 3

y = 1/3x + 1

y-intercept = 7

x-intercepts: 6, -3, -7/3

92
Q

For the parabola y = x^2 − 4x −12 in the xy-plane, find the following.

(a) The x-intercepts
(b) The y-intercept
(c) The coordinates of the vertex

A 
x-intercepts = 6, -2
y-intercept = -12
vertex = 2, -16
93
Q

For the circle ( x − 1)^2 + ( y + 1)^2 = 20 in the xy-plane, find the following.

(a) The coordinates of the center
(b) The radius
(c) The area

A 
center = (1, -1)
radius = √20
area = 20pi
94
Q

For each of the following functions, give the domain and a description of the graph y = f (x) in the xy-plane, including its shape, and the x- and y- intercepts.

(a) f(x)= −4
(b) f(x)=100 - 900x
(c) f(x)=5− (x+ 20)^2
(d) f(x)= √(x +2)
(e) f(x)=x + |x|

A

a) domain = all real numbers Looks like a horizontal line, passing through y = -4. Does not intercept x.
b) , x-intercept = 1/9. y-intercept = 100, domain all real numbers, slope of -900.
c) inverted parabola. y-intercept = -395, x-intercepts = -20 +- √5, domain all real numbers
d) half parabola, x-intercept at -2, y-intercept at √2, x >= -2
e) steep linear line starting at origin extending to the upper right. x >= 0 then the negative x-axis

95
Q

What is a domain?

A

It is the set of numbers in x.

96
Q

What is a range?

A

The set of numbers in y.

GRE Math (4 decks)

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