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1. Problem:  The sum of twice a number plus 13 is 75.  Find
             the number.
   Solution:
             The word is means equals.  The word
             and means plus.  Therefore, you can
             rewrite the problem like the following:
             The sum of twice a number and 13 equals 75.
      

             Using numbers and a variable that represents 
             something, N in this case (for number),
             you can write an equation that means the same thing
             as the original problem.
      

             2N + 13 = 75
      

             Solve this equation by isolating the variable.
      

             2N + 13 =  75     Equation.
                - 13 = -13     Add (-13) to both sides.
             -------------
             2N      =  62
             N       =  31     Divided both sides by 2.
      

2. Problem:  Find a number which decreased by 18 is 5 times its
             opposite.
   Solution:
             Again, you look for words that describe equal quantities.
             Is means equals, and decreased by
             means minus.  Also, opposite always means
             negative.  Keeping that information in mind makes it
             so an equation can be written that describes
             the problem, just like the following:
      

             N - 18 = 5(-N)        Equation.
               
             N - 18 = -5N          Multiplied out.
            5N + 18    5N + 18     Add (5N + 18) to
            ------------------     both sides.
            6N      =       18
                  N = 3            Divide both sides
                                   by 6 to isolate N.

               _          _
1. Expand: x^2| x^2   3y^3 |
           ---| --- - ---- |
           y^2|_ y      m _|
     

 Solution: Two multiplications are supposed to be done.  
           
                                      x^2     x^2
           You have to first multiply --- by --- and then
                                      y^2     y
           
                                x^2    -3y^3
           you have to multiply --- by -----.
                                y^2      m            
           
           
           That gives you:
  
           (x^2)(x^2)   (x^2)(3y^3)
           ---------- - -----------.
             (y^2)y        (y^2)m  
  
           Lastly, you simplify both expressions since 
           answers are not considered correct in Algebra
           unless they are simplified. After 
           simplification (multiply terms together and 
           cancel things out if possible), you get the 
           answer -
  
           x4   3yx2
           -- - ----
           y3     m

1. Find the LCM of 15a2b and 10ab3.

 
   Solution: Write the expressions as products 
             of prime and literal factors.
  
             15a2b                 10ab3
             3 * 5 * a * a * b     2 * 5 * a * b * b * b
  
             From the listing of prime and literal factors, 
             take the groups of factors with the most 
             instances of that factor.  For example, there 
             are three bs in 10ab3's factors, therefore, 
             three bs are listed in the LCM.
  
             The LCM is the following:
  
             2 * 3 * 5 * a * a * b * b * b = 30a2b3
    

1. Add:      a      b   
             - + -------
             x   (x + y)
     
   Solution: First, find the LCM of the denominators, 
             which will become the new denominator.  
  
             -------- + --------
             x(x + y)   x(x + y)
  
             So that the problem does not change, the 
             numerator of each term has to be multiplied 
             by the same quantity that its respective 
             denominator was.  The original denominator 
             of the first term was x, and it has 
             been multiplied by (x + y), so the
             original numerator, a, must be
             multiplied by (x + y), too.
  
             a(x + y)           
             -------- + --------
             x(x + y)   x(x + y)
  
             The original denominator of the second term 
             was (x + y), but it was multiplied by 
             x, so the original numerator must also
             be multiplied by x.  Now, the fractions
             can be added together.
  
             a(x + y)      xb      a(x + y) + xb
             -------- + -------- = -------------
             x(x + y)   x(x + y)     x(x + y)   
  
             This is one of the rare times in Algebra that 
             there are multiple forms of the correct answer.  
             In this case, you can multiply out the numerator 
             and/or denominator if you want, and since doing 
             that does not help you simplify the answer any 
             further, they are also correct forms of the 
             answer.
    

1. Simplify:  a 
              - 
              b 
             ---
              c  (b, c do not equal 0)


   Solution:  Multiply the numerator (a/b) 
              by the reciprocal of the denominator, 
              which is (1/c).


               a   1   a 
               - * - = --
               b   c   bc
    

A set is generally described as a well defined collection of distinct objects.By a well defined collection is meant a collection which is such that,given any object,we may be able to decide whether the object belongs to the collection or not.By distinct objects we mean objects objects no two of which are identical.
Elements or members of set:
The objects in a set are called its members or elements.Capital Letters A,B,C,X,Y,Z etc.,are generally used as names of sets and small letters a,b,c,d,x,y,z etc are used as members of sets.
E.g N = The set of all natural numbers = {1,2,3,...}
      W= The set of whole numbers = {0,1,2,3,...}
Ways of describing a set:There are three different ways of describing a set
1.The Descriptive Method:
A set may be described in words.for example,the set of all vowels of English alphabets.
2.The Tabular Method:
 A set may be described by listing its elements within brackets
{a,e,i,o,u}

Fraction is an expression in which we can write a number in the form P(x)/Q(x) , where Q(x) does not equal to zero.

we can check which fraction is smaller and which is greater.If the fractions has same denominator,then which fraction has greater numerator is the greater fraction.and which fraction has smaller numerator is the smaller fraction.
E.g: 4/5,6/5,3/5
In the above example 6/5 is the greatest fraction.and 3/5 is the smallest fraction.
If the fraction has different value of denominator then 1st we take two fractions,putting them equal and using cross multiplication ,then which side is greater is the greater fraction and which side is smaller is the smaller fraction.

the numbers or expressions written above the line in a fraction are called numerator

Rationalization:A fraction is to be rationalized if the denominator contains irrational numbers or complex numbers.that irrational number may be multiplied top and bottom of the fraction.

Trigonometry contains three words:
Tri means three
Goni means angle
Metron means measurement

Linear equation:an equation of the form ax+b = 0,where a doesn't equal to zero is called linear equation.
Its graph is a straight line.

Onto function:A function in which range of function is equal to the range set are called onto function.

Angle:Two rays with a common starting point form an angle.One of the ray is called initial ray and other is called terminal ray.

Circular system :A system in which we discuss about radian is called circular system.
Radian is the measure of an angle at the centre of circle by an arc whose length is equal to the radius of circle

The terms below the line of a fraction are called denominator.

there are two types of fraction:
1: Proper rational fraction:A fraction in which degree of numerator is less than the degree of denominator are called proper rational fraction.
E.g (x-1)/(x-2)(x-3)

2:Improper rational fraction:A fraction in which degree of numerator is greater than or equal to the degree of denominator are called proper rational fraction.

E.g (x-1)(x+3)/(x-2)(x-3)

3:Mixed fraction:

The sum of a whole number and a proper fraction is called mixed fraction. That sum is implied without the use of any operation such as "+"; for example, in referring to two entire biscuit and three quarters of another biscuit, the whole and fractional parts of the number are written next to each other:    .

Rational numbers are those numbers which can be written in the form of p/q where
q ≠ 0.
For example:All real numbers are also called rational numbers.
In other words" we can say that there in no gape between real numbers".
In real numbers,there is no number exist between two consecutive real numbers ,while in rational numbers there may exist infinite many numbers between two rational numbers.

A prime number which can be divisible by 1 and the number itself.  
1 is not a prime number because in the case of 1,one property is repeated.
that is why we can not say that 1 is a prime number.

   Even Numbers:the numbers which are divisible by  2 are called even numbers.Examples are : 0,2,4,6,8,... etc
odd numbers: the numbers which are not divisible by 2 are called odd numbers.