Four mangoes cost is 100 rupees
What is cost of one mango? What is unknown value
Unknown value = cost of mango
Numerical equation
If 25 is replaced by x then it is an Algebraic equation
Sum of Double of your age and half of your brother age is
equal to your father age. If your father age is 50y,then what is
your age. ??? What is unknown value
Numerical equation
If 20 is replaced by x then it is an Algebraic equation
, “5 more than a number is forty
What is unknown value
Numerical equation = 5 +35 =40
If 35 is replaced by x then it is an Algebraic equation
In Algebra numbers are represented either
by figures or by the letters of alphabets
In algebra unknown values are denoted by small alphabets,
called variables.like a,b,c,………..z
Known values or xed values like o ,1,2 ,3,4... -2,-3 …… 1/2, 2/3
3/4 ,7/4….., are called as constants
An equation is interchangeable i.e. the equation remains the
same even if LHS and RHS interchange each other.
Language of Mathematics
in Algebra
In Algebra numbers are represented either
by figures or by the letters of the alphabet.
Algebraic symbols +, — ,x, %, =
The sign + , which is read plus is placed
before a number.
a+b means, b is added to number a
The sign - which is read as minus,` is placed
before a number to indicate that it is to be
subtracted from what has gone before
@ 6-3 means that 3 is to be subtracted from 6
@ a- b + c means that b is to be subtracted from a
and then c added to the result.
a+ b -c means
d- e + f, means
-
_
Multiplication
@ The sign of multiplication is x, which is
read as ‘multiplied by ` or into
@ 6x3 means that 6 is to be multiplied by 3
@ axb means that a is to be multiplied by b
@ The sign of multiplication is generally
omitted between any two letters,
@ ab or a.b means the same as axb
@ or between a number and a letter
@ The letters are simply placed side by side
@ Sometimes the x is replaced by
a point.
@ 2 x a x b x c = 2abc = 2a .b.c
Division
@ The sign of division is ÷ which is read
‘ divided by’ or ÷
@ a ÷ b÷ c means that a is to be divided
by b and then the result divided by c
@ a÷ bxc means that a is to be divided by
b and then the result multiplied by c
@ The operation of division is often
indicated by placingtlie dividend over the
divisor with a line be tween them:
@ thus a / b is used instead of a ÷ b
Factor and Coefficient
Product And Factor
When two or more numbers are multiplied together the
result is called product ;
Each number is called a factor of the product.
Coefficient of a Term
fi
An algebraic term is written as product of two or more factors.
Each factor is
multiplying
the
remaining factors.
When the factors of a product are considered as
divided into two sets, each is called the coefficient,
that is the co factor of the other
in (6 )(abx) —> 6 is the coefficient of abx ;
(6a)(bx)—->‘6a is the coeffi- cient of bx,
(6ab)(x)——>6ab is the coefficient of x.
A numerical coefficient is defined as a fixed
number that is multiplied to a letter or
variable
In 6abx = (6 )(abx) —> 6 is the numerical coefficient of abx
Numerical Co efficient
A numerical Co efficient is number or quantity which is
attached with _ variables
It is usually an integer that is multiplied by the
variable next to it.
It magnitude or quantity of variable
It always multiplied the quantity or variables
Repeated addition always represents multiplication
Numerical co-efficient arise, by repeated addition
5 apples are repeated
5 is a number or quantity or
magnitude of apples
2 bananas are repeated
2 number or quantity or
magnitude of bananas
5 and 2 numerical Co- efficients of a and b respectively
The parts of an algebraical expression which are connected by the
signs + or - are called the terms.
Factor and Indices
When two or more numbers are multiplied together
the result is called the continued product.
Each number is called a factor of the product.
When the factors of a product are considered as divided
into two sets, each is called the coefficient, that is the co
factor of the other
in 6abx, 6 is the coefficient of abx ;
also ‘6a is the coeffi- cient of bx, and
6ab is the coefficient of x.
A numerical coefficient is defined as a fixed number that is
multiplied to a letter or variable
When a product consists of the same factor repeated any number of
times it is called a power of that factor.
Thus aa is called the second 'power of a
aaa is called the third power of a
aaaa is called the fourth power of a, and so on.
Sometimes a is called the first power of a.
Special names
aa is the square of a
aaa is cube of a,
convenient notation is ~
Instead of aa a^2 is used
instead of aaa a^3 is used
instead of aaaa a^4 is used
the factor a
being taken 4 times a ^4
instead of
aaabb
Power
Definition
m
When a product consists of the same
factor repeated any number of times
it is called a power of that factor.
Examples
Thus aa is called the second 'power of a
aaa is called the third power of a
aaaa is called the fourth power of a, and so on.
Sometimes a is called the first power of a.
Special names
aa is the square of a
aaa is cube of a,
An Algebraic Expression
collection of algebraical symbols,
that is of letters, figures, and signs, is called
an algebraical expression. Expression are either Simple or compound
An Algebraic Term
The parts of an algebraical expression which are connected
by the signs + or are called the terms.
-
Expression are either Simple or compound.
A simple expression consists of one term, as 5a
A compound
more terms.
Compression
consists
of
two
An expression of two terms, is called a binomial
An expression of three terms, is called a trinomial
When two or more quantities are multiplied
together the result is called the product.
or
Memory Note
If one or more terms combined by multiplication or
and division they treated as one term
Algebraic Like Terms
When two terms contain the same letters or
variables and which are raised to the same
power are called Like terms
Meaning of coefficient:
An algebraic term is written as
product of two or more factors. Each factor is
multiplying the remaining factors and each factor is
known as coefficient of remaining factors.
For eg. 1.
4ab: we write it in product form
4xaxb
So, coefficient of 4 = ab
coefficient of a = 4b
2. -8 p2q2 : we write it in product form
-8 x p2 x q2
So, coefficient of -8 = p2q2
coefficient of q2 = -8p2
Now we learn to separate Numerical and Literal
coefficient.
Numerical coefficient:
A numerical factor that multiplies another factor in a
term is called a Numerical coefficient. It is any
constant term that is in front of one or more variables
in a expression.
For eg.
1.
3bc:
3 x
Numerical coefficient
b x c
variables
(Constant)
2.
-2mn:
-2 x
m x n
It means 3 and -2 are numerical coefficient.
3. abc: It displays no any number, so its factor is 1.
So numerical coefficient is 1.
4. -7 x2y: It displays two numbers -7 and 2 but 2 is
an exponent and not a multiplying factor. So -7 is a
constant and it is numerical coefficient of -7 x2y.
5. – 5ab2 = It displays constant term in fraction.
3
so numerical coefficient will be =
−𝟓
𝟑
Literal coefficient:
A factor which contains at least one letter, in
product form of a term is called a literal coefficient of
the remaining factor. For eg.
1.
4a2xy: 4 x a2xy , 4 is a number and a2xy has
three letters a, x, y. So, a2xy is called Literal
coefficient.
2. -7mnp: -7 x mnp, Literal coefficient is mnp.
2
2
Sample Sums
TERM
yz
NUMERICAL
COEFFICIENT
1
LITERAL
COEFFICIENT
yz
-2abc
-2
abc
-8x2y2z2
-8
x 2y2z 2
7pqr3
6
-2ab
C
7
6
-2
pqr3
ab
c
Addition of Monomials:
! A monomial is an algebraic expression that
consists of one term. For eg. 3xy, 4abc, -7xyz.
! Two or more monomials can be added only if
they are Like Terms. Unlike terms can’t be added.
! Like Terms are terms that have exactly the same
Variable and Exponents on those variables. The
constants on like terms may be different either
positive or negative.
Constant
7 x2
Exponent
Variable
For eg. : 7xy and -8xy, 9p2q2 and -5p2q2 are like
terms
• 5x2y2 and -4x2y4 are unlike terms because
exponents are not same.
Thus
terms ;
and
are like
and
contain the same letters, but they are
not like terms, for all the letters are
not raised to the same power.
Rule I. If sum of a number of like terms is a like term.
Rule II. If all the terms are positive, add the coe�cients.
Rule III : if some terms are negative and some are positive,
then follow the steps to solve algebraic expression
a) find the total positive value
b)find the total negative value
c)finally find the difference of total positive and negative
r
values
Now we recall the integers rules of addition:
+,+
Sign of +
Add the numbers
- ,+, -
Sign of Sign of bigger
term
Add the numbers
Subtract the
numbers.
To add two or more monomials:
1.
2.
3.
Arrange the like terms in columns.
Add the numbers with integers rules.
Write the variables and their powers same.
Sample Sums
Add:
Q: 1
Sol.
7xy and 9xy
+7xy
+9xy
16xy
Q: 2
-5a2b2 , 6a2b2
Sol.
-5 a2b2
+6 a2b2
+1 a2b2
(if no sign then + sign)
Q: 3 -10 abc2 , 8abc2 , 7abc2 first add like signs
Sol. -10 abc2
+8
+8 abc2
+7
+7 abc2
+15
+5 abc2
then unlike signs
+ 15
- 10
+5
Q: 4
-7x2, x2, 4x2, -5x2
Sol.
-7x2
first add like signs
+1x2
+1
-7
+4x2
+4
-5
-5x2
+5
-12
- 7x2
then unlike signs
-12
+5
-7
Related Sums:
Add the following expressions:
1.
2.
3.
4.
5.
6.
7.
8.
7y, 8y
8abc, 3abc
-4xyz, +9xyz
6x2y2, -7x2y2
-4p3q3, -3p3q3, +5p3q3
2a2b4, -4a2b4, 7a2b4
-3x, -5x, 6x, 9x
6ab, -8ab, -3ab, 4ab
Class 7th
Subject: Maths
Topic: Addition of binomials
(Repeated topic of class 6th)
Recap (Worksheet 3)
Addition of Binomials.
• A binomial is an algebraic expression that
consists of two terms. For eg. 5x + 3z, -4a2+ 6b2
• Two or more terms can be added only if they are
Like Terms.
Now we recall the integers rules of addition:
+,+
Sign of +
Add the numbers
- ,+, -
Sign of Sign of bigger
term
Add the numbers
Subtract the
numbers.
Now unlike signs
-11
+11
+9
-2
-2
+9
Related Sums
Add the following expressions:
1.
2.
3.
4.
7a + 9b, 8a -4b
-3x + 5y, 6x – 10y
8p – 3q, -7p -2q, -9p + 4q
10a -2b, +7a +4b, -3a + 6b
What is the difference in meaning between 3a and
The expressions abc ——>, abc, bca, cab, cab, acb, bac have
the same value,
ईच denoting the product of the three quantities a, b, c.
If x=0 then
Therefore
contains a zero factor.
=0 when x=0, whatever be the values of
a,b,y
Any term which contains a zero factor is itself zero, and may
be called a zero term
3x4 = 4x3.
In a similar way, 3x4x5=4x3x6=4x5x3;
The factor ‘ a’ being taken n times a^n
a^n = a x a x a x a x a x a x a……. n’times
repeated multiplication
The quantity which when squared is equal to any number a is called
the square root of a, and is represented by the symbol
The quantity which when cubed is equal to any number a is called the cube root
ofa,and
is
represented
by
the symbol
.Indices practice
In general, the quantity which when raised to the nth power, where n
is any whole number, is in radical form and is represented by the
symbol_____.
The sign
It is often called the
N l sign
A root which cannot be obtained exactly is called a surd, or an irrational
quantity
The approximate value of a surd, for example of root seven, can be found, which
may be by the ordinary arithmetical process
No approximate value of
1 Identity for Addition
0 is the identity element for addition
because 0 + 5 = 5 AND 5 + 0 = 5.
0 + a = a and a + 0 = a
2 Identity for Multiplication
1 is the identity element for multiplication
because 1 · 5 = 5
(1)a = a and a(1) = a
3 Additive Inverses
Additive inverse are opposite numbers
The opposite of −3 is 3. Thus −(−3) = 3.
Sum of two additive inverse two numbers is zero
4 Multiplicative inverse
Product of two multiplicative inverse is one
Product of the reciprocal is also one
In general, a · 1.
a
=
1