Class 6 – Algebra

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Algebra
  • Introduction to Algebra
  • Addition and Subtraction
  • Multiplication
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Algebra
  • Simple equations
  • Simplification of Brackets
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Introduction: In arithmetic, we deal with specific numbers and use the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 to represent them. We use the four fundamental operations namely, addition (+), subtraction (-), multiplication (×) and division (/) on specific numbers and obtain numerical expression.

Numerical Expression:
A combination of numbers connected by one or more symbols +, –, × and / is called a numerical expression.

Example: 14 – 2 × 3 + 18 / 6 is a numerical expression.

Algebra: Algebra is a generalized arithmetic in which numbers are represented by letters, known as literal numbers or simply literals.

Constants and Variables: There are two types of symbols in Mathematics, namely constants and variables.

Constants: A symbol having a fixed value is called a constant.

Thus each of the symbols 3, -5,  etc., is a constant.

Variables: A symbol which can be assigned various values is called a variable.

Example: We know that the diameter of a circle is twice its radius.

Diameter = 2 × Radius

d = 2 × r

Here 2 is a constant

These values of r and d are not fixed

We may give any value to r and get the corresponding value of d.

When r = 4 we get d = 2 × 4 = 8

When r = 3 we get d = 2 × 3 = 6 and so on

So r and d are variables.

Thus, r and d represent numbers, variables are also known as literals.

We use small letters a, b, c, x, y, z etc., to denote literals.

Operations on literals and numbers: Since literals represent numbers, they follow all the rules of addition, subtraction, multiplication and division.

 

Addition of literals and numbers:

i)     10 added to a is written as
a
+ 10

ii)   y added to x is written as x + y

 

The properties of addition of numbers also hold good in algebra:

For any literals x, y, z we have.

x + y= y + z (commutative law)

(x +y) + z = x + (y + z) (Associative law)

x + 0= 0 + x = x (additive identity)

 

Subtraction of literals and numbers:

i)    p decreased by 5 is written as  p – 5

ii)  y subtracted from 7 is written as 7 – y

iii) x decreased by z is written as x – z

Multiplication of literals and numbers:

i)    5 times b is written as 5 × b denoted by 5b (not b5).

ii)  The product of x and y = x × y, written xy.

iii) 1× x is simply written as x and not 1x.

iv)  (-1)× x is written as –x.

 

The properties of multiplication of numbers also hold good in algebra.

xy =yx(Commutative law)

(xy)z = x(yz) (Associative law)

x × 1= 1 × x = x  (Multiplicative identity)

x(y +z)  = xy + xz (Distributive law)

x × 0= 0 × x = 0

 

 

 

Division of literal and numbers:

i)    8 divided by b is written as

ii)  a divided by 4 iswritten as

iii) y divided by zis written as

Examples: Write the following in mathematical form using signs and symbols.

i)    3 decreased by a.

ii)  c taken away from a.

iii) 5 times a added to 7 times b.

iv)  One third of b multiplied by the sum of a and 8.

v)   5 less than the quotient of b by 10

vi)  a increased by 4.

Solution: We may write the above as:

i)    3 – a

ii)  a – c

iii) 5a +7b

iv)  b(a+8)

v)   

vi)  a + 4

Example: Write each of the following statements in mathematical form using signs and symbols:

i)    4 times a increased by 5 gives 8.

ii)  20 decreased by x equals 3

iii) a exceeds b by 15

iv)  6 less than quotient of x by 5 equals 2.

v)   7 times x is greater than 18

vi)  Quotientof 7 by n is less than 9

Solution:

i)    4a+ 5 = 8

ii)  20– x = 3

iii) a = b+ 15

iv) 

v)   7x> 18

vi) 

 

Example: Write the following in exponential form:

i)    p × p × p ———-8 times.

ii)  6× a × a × c × c × c.

Solution:

i)    p × p × p × ———-8 times = p8.

ii)  6× a × a × c × c × c = 6a2 c3.

 

Example: Write  inproduct form.

Solution:  =

 

Algebraic Expression: A combination of literals and numbers connected by one or more of the symbols +,–, × and ¸ is called an algebraic expression.

The several parts of an algebraic expression separated by + or signare called the terms of the expression.

Examples:

i)    2x + 3y – 5z is an expression having 3 terms, namely 2x, 3y and 5z

ii)  x2 – 4y2z + 3zx– 9 = 0 is an expression having four terms namely
x2, –4y2z, 3zx and 9 constant term.

 

Various types of Algebraic expression:

i)    Monomial: An algebraic expression having only one term is called a monomial.

Examples: 5b, – 4a, 6ab, 7,  –9, ,  is monomial.

ii)  Binomial: An algebraic expression having two terms is called a binomial.
Examples:

a)   5x– 3 is a binomial having 2 terms namely 5x  and –3

b)   6 –2y is a binomial having 2 terms, namely 6 and –2y.

iii) Trinomial: An algebraic expression having three terms is called a trinomial.
Example: 3x + 4y – 5 is a trinomial having 3 terms, namely 3x, 4y, -5

iv)  Multinomial: An algebraic expression having more than 3 terms is called a multinomial.
Example: a2+b22ab + 8 is a multinomial having 4 terms, namely, a2,b2, -2ab and 8.

Factors of a term: When numbers and literals are multiplied to form a product, then each quantity multiplied is called a factorof the product.

A constant factor is called a numerical factor while a variable factor is aliteral factor.

Examples:

i)    In5xy: The numerical factor is 5

The literal factors are x, y and xy.

ii)   In–3a2b: The numerical factor is –3

The literal factors are a, a2, b, ab and a2b

Constant term: A term of the expression having no literal factor is called the constant term.

Example:

i)    In the expression x – 2y + 7, Constant term is 7.

ii)  In the expression x2 + y2 – xy – 5, the constant termis – 5.

Coefficients: Any of the factors of a term is called the coefficient of the product of other factors. In particular the constant part is called the numerical coefficient of the term and the remaining part is called the literal coefficient of the term.

Example:

i)    Consider the term 8a2b numerical coefficient = 8

Literal coefficient of the term = a2b

Also,the coefficient of b = 8a2

And,the coefficient of a2 = 8b

ii)  Considerthe term – abc2

Numerical coefficient of the term = -1

Literal coefficient of the term = abc2

The coefficient of a = –bc2

The coefficient of b= –ac2

The coefficient of c2 = -ab

iii) Consider the term p2q

Numerical coefficient of the term = 1

Literal coefficient of the term = p2q

The coefficient of p2 = q

The coefficient of q = p2

Like terms: Two monomials having same literal factors are called like or similar terms.

Unlike terms: Two monomials not having same literal factors are called unlike or dissimilarterms.

Example:

i)    6ab,–3ab, 7ab are like terms.

ii)  –8ab2,7ab2,  are liketerms.

iii) 9x,9xy are unlike terms.

iv)  2xy2,2x2y are unlike terms.

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For example, if x = 2 and y = 3. Lets evaluate the original expression and the two rearrangements and make an observation.

We observe that through arrangement is different result insame for evaluating an expression.