Learning Linear Equations in Two Variables

In this edition we will study about Linear Equation in Two Variables. First of all we need to be clear on what is meant by a Linear Equation.

Linear Equation: An equation which represents Linear Graph is called as Linear Equation. Is is an equation in which each term is a constant or the product of the constant. It can have one or more Variables.

Further let us see the various forms of solving Linear Equations namely:-

1. Graphical Method
2. Algebraic Method

Under Algebraic method there are three methods namely:-

1. Elimination by substitution method
2. Elimination by equating co-efficient method
3. Cross multiplication method

Will come next with a brief description on all the above............... To be continued................

Limits of a Sequence

The limits of a sequence is one of the important concepts of mathematics. The sequence or a series when tends to converge at a point then that point is called as the limit. Let us consider that we are having a sequence of points and a point L is said to be the limit of the sequence. In this article we are going to see about the limit of a sequence.

Definition of Limit of a Sequence:

Let us consider a series of real numbers {xn │n € N}, where the real number L is called as the limit of the sequence xn such that

lim xn = L
n->∞

which is true for every real number ε > 0 and there appears the number N such that for every number n>N,there exists │xn - L│ < ε.

Let us consider that for a series of points that has been given by { xn │n € N }with the distance function f. Then the element L € M is called as the limit of the sequence which is given by lim xn = L, which is true for every real number ε > 0 and there appears the number N such
n->∞

that for every number n>N, there exists (f(xn), L) < ε.

Let us consider an element L € T as the limit of the sequence only if for neighborhood N of L, there exists a number N when xn € N for all the values of n > N.

If a sequence contains a limit then the sequence is said to convergent and it converges up to the limit, else the sequence is said to be the divergent sequence. A null sequence does not contain any limit and it converges to zero.

Properties of Logarithm

Below are some properties of Logarithmic algebra of positive numbers.

(i) Product rule: If a, m and n are positive numbers and a ≠1, then

log_a (mn) = log_a m + log_a n

(ii) Quotient rule: If m, n and a are positive numbers and a ≠ 1, then,

log _a(m/n) = log_a m - log_a n

(iii) Power rule: If a and m are positive numbers, a ≠ 1 and n is a real number, then

log_a mⁿ = n log_a m

(iv) Change of base rule: If m, n and p are positive numbers and n ≠ 1, p ≠ 1, then

log_n m = (log _p m) (log_np)

(v) Reciprocal rule: If m and n are positive numbers other than 1, then

log_nm = 1/(log_mn)

(vi) If a is a positive number, then log_a 1 = 0

(vii) If a is a positive number, then log _aa = 1

(viii) If a and b are any two positive numbers and b ≠ 1, then b log_b a = a

(ix) Let m, n and a be positive numbers and a ≠ 1.log am = log an = then m = n.

Percentiles

The percentiles are finding out the certain percent of observation fall. The percentiles rank can be found out with the help of following formula.

Percentile Rank = ((B + 0.5E)/n) * 100

Where,

B = number of scores below x E = number of scores equal to x n = number of scores

• The Percentiles measure helps us to find out what percent of the total frequency was scored at or below the given measure. The percentiles rank is the percentage of scores that falls at or below a given scores.
  
• Percentiles are the easiest method to understand, when we look at the comparison of the percentage and the percentiles.
  
• When we are looking over the set of data, percentiles helps a lot in middle or median performance, or any type of the data.
  
• Percentiles are used in testing the population group when they are taken for a survey. Once again we repeating that comparison of scores are the percentiles.

Remainder Theorem

Remainder Theorem: If a polynomial f ( x ) is divided by ( x - a ), then the remainder = f ( a )

Let p ( x ) be any polynomial of degree greater than or equal to one and let a be any real number. If p ( x) is divided by a linear polynomial x - a , then the remainder is p ( a ).

Proof for Remainder theorem in Algebra

Proof: Suppose when p ( x ) is divided by x - a , the quotient is q ( x ) and the remainder is r ( x ).
Then p ( x ) = ( x - a ) q ( x ) + r ( x )
where degree r ( x ) <>
Since, degree ( x -a ) = 1, therefore r ( x ) is a constant , say r
For all values of x,
p ( x ) = ( x - a ) q ( x ) + r
for x = a ,
p ( a ) = ( a - a ) q ( a ) + r
= 0 x q ( a ) + r
= r
Hence the theorem is proved.