Mathematics
"Mathematics rightly viewed, possesses not only truth, but supreme beauty; a beauty cold and austere, like that of sculpture, without appeal to any part of your weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show."
Straight
The Cartesian plane is intended to represent a point by coordinates and consists of two number lines perpendicular; the horizontal axis is called the abscissa (x) and the vertical axis ordinate (y).
The line represents relationships between quantities and one of its features is the inclination also called slope (m); let (x1, y1) and (x2, y2) two different points on a vertical line not the slope of the line will be
Formulates
Y2  Y1
X2  X1
m =
Y  Y1 = M (X  X1)
Y= m x + b
m = slope
b = where the line height
Variants
two points
slope point
equation of the line
Ax + By + C = 0
general equation of the line
i = p q
CT = Cvq + CF
U = IT  CT
utilities
ingress
total cost
1) Customers demand 40 units of a product when the price is $ 12 by unit and 25 units when te price is $18. Set the price by unit to 30 units.
18 12
25  40
Y2  Y1
X2  X1
m =
6
15
=
=
=
0.4
Y  Y1 = M (X  X1)
Y 12 = 0.40 (X  40)
Y= 0.40X + 16 + 12
Y = 0.4X + 28
=
Y= m x + b
1
2
0.40X + 28 = 0
X= 28 / 0.4 = 70
3
A (0, 28) B (70,0)
4
P= 0.4 (30) + 28
P= 12 + 28 = 16
5
(0,28)
(70,0)
m =  0.4
2) A manufacturer of refrigerators produced 3,000 units when the price is $ 940 and 2200 units when the price is $ 740. determined the supply equation.
Y2  Y1
X2  X1
m =
740  940
2200  3000
200
800
=
=
=
0.25
Y  940 = 0.25 (X  3000)
Y  940 = 0.25X  750
Y = 0.25X + 190
0.25 + 190 = 0
X = 190 / .25 = 760
A (0, 90) B(760, 0)
1
2
3
4
Linear equations
An equation is a proposition stating that two expressions are equal, and is formed by members or sides and an equal sign (=) (A=8). Each equation has at least one variable. A variable is a symbol (z, y, x) that can be replaced by a number.
Solve an equation means finding the values of the variables or unknow.
Examples
1) 2 (x  4) = 7
10
2 (x  4) = 7 (10)
2x  8 = 70 2x= 62 x= 62/2 x =31
2) 2x  3 = 6
4x  5
2x  3 = 6 (4x  5 )
3
2
2
2x  3 = 24x  30 2x  24x = 30 + 3

22x = 27 x= 27/22 x = 1.22
System of equations
How to describe mathematical situations resulted in the systems of equations with numerical coefficients characterized by their relative position, and can be described by the rectangular array.
It used to find the point where two lines intersect and there are several methods to achieve them:
Gass – Jordan (elementary row operations)
1) can be exchanged lines
2) multiply a whole row or row by a constant
3) for the zeros can multiply a row by a constant and the resulting adds a row where you need to zero
Equalization
The same variable is cleared in both equations and clearances are equal
Replacement
One variable is cleared in a clearing equation and is replaced in the other, then resolves
Add and subtract
One of the variables by adding or subtracting member by member, an equation or a multiple of this to find values is eliminated.
Graphic method
The graphs are plotted or tabulated the two equations.
Crammer (determinants)
The number of equations equals the number of unknowns.
1) The determining coefficients are the unknowns without result and what the relationship is divided with other unknowns.
2) The relationship to other unknowns results from the substitution of factor of question that you want to find by the result.
3) It multiplies cross and the sign is exchanged in each column.
Condition inequalities or inequalities
Sentence which states that a number is greater or less than another; with one or more unknowns.
Members:
It is called the first member of an inequality expression that is on the left and a second member which is to the right of the inequality sign. A>B A+3 >B+3
Characteristic
If the two members are multiplied or divided by the same positive number, the sign does not change.
AC>BC A/C> A/B
If the two members are multiplied or divided by the same negative number, the sign of the inequality varies.
A>B A –C< BC A/C < B/C
Any member may be replaced by an equivalent expression to it.
A>B and A=C now C>B
If two members are reverted, inequality change sign
A>B 1/A <1/B
If the two sides of the inequality are positive and each side rises by the same positive potency does not change sign.
6^(2 )> 3^2 36>9 or √(2&36 ) > √(2&9)
A company offers a position in sales and you can choose the method of payment of his salary. the first pay $ 12.600 plus a bonus of 2% of annual sales. the second pays a direct commission of 8% of sales. What level of sales you should be the first method?
1) 12600 + 2%(V) 2) 8% (V)
12600 + .02 (V) > .08 (V)
12600> .08.02 (V)
12600> .6 (V)
12600/.06> V
V <210,000 when you sale more than 210,000
The rapid rate of business is why liquidity of its assets less cash, securities and accounts receivable from their existing obligations. If a company has $ 450,000 in cash and securities and has $ 398,000 in bonds. As need to consider receivable reason to keep above 1.3?
450000 + X
398000
450000 + X > 1.3 (398000)
A59000 + X > 517400
X>517400  450000
X> 67400
> 1.3
Cuadratic equation
It is the variable "x" in an equation that can be written ax^2+ bx + c = 0 ; where a, b, and c are constants it is different from zero
pure
ax^2+ c = 0
mixed
ax^2+ bx = 0
complete
ax^2+ bx + c = 0
Pure
1) x^2  16 =0 x^2+ 0 + 16 = 0
factorize (x4)(x+4)
2) x^2  4 = 0 x^2 +0 4 = 0
factorize (x2) (x+2)
Mixed
1) x^2  8x = 0 x=  (8) + √(8)^2  4(1)(0)
2
x1= 8+8 /2 = 8 x2 = 8  8 / 2= 0
2) 2x^2 + 4x = 0 x=  (4) + √(4)^2  4(2)(0)
4
x1= 4 +4 /4 = 0 x2= 4  4 /4= 2
Complete
1) X^2 + 9X + 14 = 0 X=(9) + √(9)^2  4(1)(14)
2) X^2  13X + 36 = 0 X=(13) + √(13)^2  4(1)(36)
2
2
X=(9) + √81 56 X1= 9 + 5 /2 = 2
X2= 9  5 /2 = 7
X=(13) + √169 144 X1= 13 + 5 /2 = 9
X2= 13  5 /2 = 4
2
2
Potency
It is the same algebraic or numerical expression or the result of taking it as a factor two or more times.
1) every potency even in a negative quantity is positive
2) all odd potency of a negative quantity is negative
Examples
Algebra
Numbers and letters are used as symbols where numbers are used to represent known quantities and unknown quantities letters; to represent quantities which use algebraic formulas are used three types:
Signs of operation (addition, subtraction, division, multiplication, potency and root)
Relationship signs (equal =, greater [ ], than >, less than <)
Grouping symbols (parenthesis (), clasp and keys {}
Rulers factor
Examples
1) (8X  4 + 2) + (3X + 2Y  5) = 11X  2Y +7
2) (2X + 3) (5X + 2) = 10X^2 +19X +20
ordinary factot
perfect square trinomial
difference square
addition for two cubes
difference for two cubes
Example
Examples
1
X + 2Y = 1
4X + 5Y = 3.25
1 2 1
4 5 3.25
 4 R1 = 4 8 4
4 5 3.25
0 3 .75
1 2 1
1 3 .75
1/3 R2 0 1 .25 NR2
1 2 1
o 1 .25
 2 (RI) 0 2 .5
1 2 .1
1 0 .5 NR1
1 0 .5 > X
0 1 .25 > Y
X + 2Y = 1
4X + 5Y = 3.25
X + 2Y = 1
4X + 5Y = 3.25
1 2
4 5
D=
58 = 3
1 2
3.25 5
1 1
A 3.25
X
X=
Y=
5 6.5 = 1.5
3.25 4 = .75
X= 1.5/3= .5
Y= .75/3= .25
(4) = 4X  8Y = 4
4X + 5Y = 3.25
Y= .75/3 = .25
(.25) = .25X  5Y =2.5
4X + 5Y = 3.25
X= .75/1.5= .5
(7*6) ^3= 7^3 * 6^3= 343 * 216 = 74088
7^5 / 7^6 = 1/7
Law Examples
5x  3y = 1
8x + 2y =12
x=0 Y=12/2 (0,6)
y=0 x=12/8 ( 2/3, 0)
2a + c = 1
4.9 + 2.2c = 2.40
c= 12a
4.9a + 2.2 (12a) = 2.40
4.9a  4.4a = 2.40  2.2
a = .20/.5 = .40
c= 1 (2+.40)
c=.20
2a + c = 1
4.9 + 2.2c = 2.40
c= 2a + 1
c = 4.9/2.2 a + 2.40/2.2
2.2(2a+1) = 4.9 + 2.4
4.4a +2.2 = 4.9a +2.4
.5a =.2
a=.4
(.4)2 + 1 =c = .20
1) x+ (44*11)= 567
x = 567 484
x = 83
2) m + {5 (34+ 76)} = 789
m + 550 = 789
m= 789550 = 239
m=239
Bertrand Russell
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