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CBSE-i
mathematics
Algebraic Expressions
Student's
Section
Preet Vihar,Delhi-110 092 India
CBSE-i
mathematics
Algebraic Expressions
Preet Vihar,Delhi-110 092 India
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1. Study Material
2. Student's Support Material (Student's worksheets)
C
SW 1: Warm Up Activity W1
1
32
33
Variables and Constants
C
SW 2: Warm Up Activity W2
34
Forming Expressions
C
SW 3: Pre Content Worksheet P1
37
Types of Algebraic Expressions
C
SW 4: Pre Content Worksheet P2
38
Degree, Terms and Coefficients
C
SW5: Content Worksheet C1
41
Algebraic Expressions Through Pattern
C
SW 6: Content Worksheet C2
44
Like and Unlike Terms
C
SW 7: Content Worksheet C3
46
Addition of Algebraic Expressions
C
SW 8: Content Worksheet C4
48
Subtraction of Algebraic Expressions
C
SW 9: Content Worksheet C5
50
Skill Drill
C
SW 10: Content Worksheet C6
52
Addition and Subtraction of Algebraic Expressions
C
SW 11: Content Worksheet C7
Multiplying Monomial with Monomial
54
C
SW 12: Content Worksheet C8
56
Multiplying a Monomial with a Binomial
C
SW 13: Content Worksheet C9
58
Multiplication of Binomial Algebraic Expressions
C
SW 14: Content Worksheet C10
59
Evaluate Algebraic Expressions
C
SW 15: Content Worksheet C11
61
Expression, Equation, Formula and Identity
C
SW 16: Content Worksheet C12
64
Simple Linear Equations
C
SW 17: Content Worksheet C13
68
Solving Simple Linear Equations
C
SW 18: Content Worksheet C14
70
Word problems
C
SW 19: Content Worksheet C15
74
Simple Equations with Variables on Both Sides
C
SW 20: Post Content Worksheet PC1
77
Independent Practice
C
SW 21: Post Content Worksheet PC2
80
Test Your Progress
Suggested Videos/ Links/ PPT's
88
STUDY
MATERIAL
1
ALGEBRAIC EXPRESSIONS
Introduction
In Class VI, you were introduced to the concepts of variables and constants. You have
also learnt how to form expressions called algebraic expressions using variables and
constants by using fundamental operations (+ , - , x , ). In this unit, we shall first
recapitulate these concepts and study more about algebraic expressions.
Addition and subtraction of algebraic expressions will also be discussed along with
simple cases of multiplication. Further, we will introduce simple equations and learn to
solve and apply them in daily life.
1. Formation of Algebraic Expressions : A Review
Variable : A quantity that can take different values is called a variable. Infact, the
word ‘variable’ means which can vary. The value of a variable is not fixed. They
are usually denoted by letters x, y, z, , m, n, etc. However, these variables
denotes same numbers only that is why they are also called literal numbers or
simply literals.
Constant : A quantity which has a fixed value is called a constant. The numbers
2, 9, 100, , etc., are constants as they have fixed values. For example, number of
vertices of a triangle is 3 which is a constant.
Sometimes constant are also denoted by letters a, b, c, etc.
Algebraic Expressions:
Recall that variables are also numbers. So, all the fundamental operations
(addition, subtraction, multiplication, division) can be performed on them also as
in case of numbers. The expressions obtained by performing different
fundamental operations on variables and constants are called algebraic
expressions.
For example:
y + 2, is an algebraic expression formed by ‘adding 2 to y’,
15x, is an algebraic expression formed by ‘x multiplied by 15’
2x—3 formed by ‘first multiply x by 2, then subtract 3 from the product’
, is an algebraic expression formed by, ‘y divided by 9’
-5t , is an algebraic expression formed by , ‘t multiplied by -5’, etc.
2
Similarly,
2x + , 10p, 2n – 11, 5y + 10, + 3x
8y – 3 , 2x + 19, etc., are also algebraic expressions.
In the algebraic expressions:
2x + , 2x and are terms.
10 p , 10p is a term.
-5y + 10 , -5y and 10 and are terms.
8y – 3, 8y and -3 are terms.
5 – 7x, 5 and -7x are terms.
Recall that in the term
2x,
,
2 is constant and x is a variable
is a constant
10 p,
10 is a constant and p is a variable
5y,
5 is a constant and y is a variable
7x,
7 is a constant and x is a variable
3,
3 is a constant
and so on.
Recall that in a term, constant is called a numerical coefficient of the variable.
For example in the term
2x, constant ‘2’ is a numerical coefficient of x
-7x , constant’ -7’ is a numerical coefficient of x
etc. Further in the term 2x, 2 is also called coefficient of the term 2x. Similarly, in
the term -7x ‘-7’ is coefficient of the term 7x and so on.
3
2. Generating More Algebraic Expressions
Consider the following situation:
(i)
Length of a rectangle is x and breadth is y. Its perimeter = 2x + 2y
(ii)
Area of the rectangle = x x y, also written as xy.
(iii)
If side of a square is x, then
its perimeter = 4x,
Area of the square = x x x, written as x2.
As 2 x 2 =
( 3) x ( 3) = ( 3)2,
Similarly
x x x = x2
x2 is read as the second power of x or square of x or x raised to the exponent 2 or
simply x raised to 2.
In the same way
x x x x x is written as x3, read as the third power of x, or cube of x or x raised to 3;
and
x x x x x x x is written as x4, read as fourth power of x, or x raised to 4 and so on.
x x x x x x ….. x x (n times)
is written as xn and read as nth power of x or x raised to n,
where n is natural number.
n is called exponent and x, the base.
In the same way, we have expressions like
x x x x y, written as x2y.
x x y x z, written as xyz.
3 x x x y , written as 3xy
4
(-8) x x x x x y, written as 8 x x2 y = 8x2y and so on.
The above expressions are also examples of algebraic expressions.
These expressions are obtained by performing fundamental operations on variables and
constants.
Similarly, we can have more algebraic expression like
7a2 b2, x10, x2 y5, 5x3 y3 z3 , x2 –y2,
x2 + y2, 5x2 + 9, 5x3 + y2 – z, 3x2 2y2,
4mn2 – 7mn + 4n m2, etc.
Term of an Algebraic Expression
The algebraic expression:
x2 – y2 has two terms x2, -y2
7a2b2 has only one term 7a2b2
x2 + y2 has two terms x2, y2
5x2 + 9 has two terms 5x2, 9
5x3y3 z3 has only one term 5x3 y3 z3
3x2 – 2y2 has two terms 3x2, -2y2
4mn2 – 7mn + 4nm2 has three terms 4mn2, 7mn and 4nm2
and so on.
Algebraic Expressions having only one term are called monomials, having two terms,
are called binomials, having three terms are called trinomials.
Thus, 5 x3 y3 z3 is a monomial.
Similarly, 5p2, -9y2, 11, etc., are also monomials.
and so on.
x + y,
a+b, a2 + b2, 3x + 4z, 5x2 + 9, etc are binomials.
x + y + z, 4mn2 – 7mn + 4nm2, etc., are trinomials.
Coefficient of a Term
5
A term has two parts (factors). One is numerical and the other is algebraic
(or literal) For example:
In 7a2 b2, 7 is the numerical part and a2 b2 is the algebraic part In x2 y5, + 1 (or 1)
is the numerical part and x2 y5 is the algebraic part,
In – 5x3, 5 is the numerical part and x3 is the algebraic part.
and so on.
The numerical part in a term is called its numerical coefficient or simply
coefficient.
For example, in the term 7 a2 b2, 7 is the numerical coefficient or simply the
coefficient of the term 7 a2 b2.
It is also the coefficient of a2 b2.
In the term x2 y5, + 1 (or 1) is the numerical coefficient (or coefficient) of the term
x2 y5.
and + 1 also coefficient of x2 y5.
In the term, 5x3, 5 is the coefficient of the term as well as of x3.
In the term, –y2z, 1 is the coefficient of the term as well as of y2z
Note that
(i)
Sign of the term is always included in the coefficient.
In the terms like x2y5 and y2, the coefficients are + 1 and 1 respectively.
Example 1: Indentify constants, variables and coefficients of each term in the following
algebraic expressions:
(i)
2x2 – y
(ii)
5a2 b2 – 6b2c2 + 3
(iii)
x3 – 7x2y – 5xy2 + 1
6
Solution :
Expression
(i)
(ii)
2x2 – y
5a2b2 – 6b2c2 +3
Terms
Constants
Variables
Coefficients
2x2,
2
x
2
y
1
y
1
5
a,b
5
6b2c2,
6
b,c
6
3
3
–
–
x3
1
x
1
7x2y
7
x,y
7
5xy2
5
x,y
5
5a2b2,
(iii) x3-7x2y-5xy2 +1
1
1
–
Powers of Variable in a Term
In the algebraic expression:
2x2 y, the term 2x2 has variable x, exponent of x is 2 and the power is x2.
The term –y has variable y, exponent 1 (y can be written as y1) and power is y
In the algebraic expression 7a2b2,
there is only one term, variables are a and b, exponent of a is 2 and its power is
a2, exponent of b is 2 and its power is b2 and so on.
Example 2: Write down the variable, exponent and power in each of the following term:
(i)
2x2y
(ii)
(ii)
y2x3t4
(iv)
4 x3 y2 z
3 a2 b2 c2
7
Term
(i)
(ii)
2x2y
–4x3y2z
(iii) y2 x3 t4
(iv) 3a2 b2 c2
Variables
Exponent
Power
x,
2
x2
y
1
y
x,
3
x3
y,
2
y2
z
1
z
y,
2
y2
x,
3
x3
t
4
t4
a
2
a2
b
2
b2
c
2
c2
Degree of a term
In the term 2x2, there is only one variable x.
Its exponent is 2. We say that degree of the term 2x2 is 2
In term, y, there is again only one variable y.
Its exponent is 1. We say that degree of the term –y is 1.
Similarly, degree of the term 7x3 is 3 and that of 8y7 is 7 and so on.
In the term 2x2y, there are two variables x and y. Exponent of x is 2 and exponent of
y is 1.
We say that degree of the term 2x2y is (exponent of x + exponent of y), i.e., 2 + 1 = 3
Similarly, degree of the term 4x3 y2 is (exponent of x + exponent of y), i.e., 3+2 = 5,
and degree of the term y2 x3 t is 2+3+1 = 6
degree of the term 3a2 b2 c2 is 2+2+2 = 6
degree of the term –8m2 n3 p4 is 2+3+4 = 9 and so on.
8
Example 3: Find the degree of the following terms:
(i) 8x4y
(iv) m2 n
x2yz2
(ii)
3
(v)
(iii)
xyz
15y10
Solution :
(i)
Degree of 8x4 y is 4+1 = 5
(ii)
Degree of –x2yz2 is 2+1+2 = 5
(iii)
Degree of xyz is 1+1+1 = 3
(iv)
Degree of m2n
(v)
Degree of 15y10 is 10
3
is 2+1+3 = 6
What is the degree of a constant, say 5?
5 can be written as 5xo (Recall ao = 1 for any non-zero a)
So, degree of 5 is 0.
Similarly, degree of -5 is also 0.
Degree of 7x is 1 but degree of 7 is 0.
Degree of 100 is again 0 and so on.
Thus, degree of a non-zero constant is always 0.
Degree of an Algebraic Expression
Consider the algebraic expression:
2x2 – y2 + z3 + xy
Here terms are: 2x2,
Degrees of term : 2,
y2, z3 and x y
2,
3,
2, = (1+1)
Among 2, 2, 3 and 2; 3 is the highest
We say that 3 is the degree of the algebraic expression.
Similarly, in the algebraic expression
8x2y – 6xy2 – 2xy – 7 z2x + 7x, we have
Degree of 8x2y = 2 + 1 = 3,
9
Degree of 6xy2 = 1 + 2 = 3
Degree of 2xy = 1 + 1 = 2
Degree of z2x = 2 + 1 = 3
Degree of 7x = 1
So, degree of the algebraic expression
= highest of 3, 3, 2, 3, 1
=3
In the algebraic expression 9x + 5
Degree of 9x = 1
Degree of 5 = 0
So, the degree of algebraic expression is 1
In general, degree of an algebraic expression is the highest degree among all its
terms
Example 4: Find the degree of each algebraic expression
(i)
x3 + y3 + 9y2
(ii)
ab + b + 5
(iii)
a2 + b2 + c
(iv)
2xy
(v)
7x
15x –y2
8
Solution:
(i)
In the algebraic expression x3 + y3 + 9y2,
degree of x3 = 3
degree of y3 = 3
degree of 9y2 = 2
10
So, degree of the given expression is 3
(ii)
Degree of algebraic expression ab + b + 5 is 2 (Why?)
(iii)
Degree of a2 + b2 + c is 2.
(iv)
Degree of 2xy – 15x –y2 is 2.
(v)
Degree of 7x– 8 is 1.
3. Like and Unlike Terms
Consider the following pairs of term:
(i)
(ii)
8x, 5x
7y, 6y
(iii)
8a2b,
8ab
(iv)
5x2y, yx2
(v)
7x, 7y
(vi)
2xy, 6xy
(vii)
5xy, 10x
Examine the algebraic part of terms in each pair.
We have:
(i)
x, x
(ii)
y, y
(iii)
a2b, ab
(iv)
x2y, yx2
(v)
x, y
(vi)
xy, xy
(vii)
xy, x
In (i) , (ii) and (vi) algebraic parts of both terms are the same.
We say that these terms are like terms.
In other pairs, algebraic parts are not the same.
11
We say that these terms are unlike terms
Thus, two (or more) terms having the same algebraic parts are called like terms
When the term have different algebraic parts they are called unlike terms
Example 5: Indentify like terms in the following:
(i)
a2, b2, 2a2, c2
(ii)
2mn, mp, 3m, 3pm
(iii)
3yx2, x2z , x2y, x2y2
(iv)
abc2, ab2c, 5a2b, -ac2b, acb2, ba2c
(v)
pq2, 8q2p, p2q, pq, 3p2q, 4q2p
Solution:
(i)
Like terms are : a2, 2a2
(ii)
Like terms are: –mp, 3pm
(iii)
Like terms are : 3yx2, x2y
(iv)
Like terms : abc2, ac2b; ab2c, acb2 and
5a2bc, ba2c
(v)
Like terms are: pq2, 8q2p, 4q2p; p2q,
3p2q
4. Operations on Algebraic Expressions
Recall that an algebraic expression may contain like terms and unlike terms. In
adding or subtracting algebraic expressions, we add or subtract like terms after
combining them together. Let us see how do we add or subtract like terms.
Additions and subtraction of like terms
Consider the terms 9x and 7x.
We can add them as follows:
9x + 7x = (9+7) x
=16 x
[Recall that 9x and 7x are also numbers. So, we
can add them using distributive property]
Similarly, 11a + 3a = (11 + 3) a
= 14 a
12
5pq + 9pq = (5+9) pq = 14pq
m2n + ( 5m2n) = [1+( 5)] m2n = 4 m2n
Thus,
The sum of two like terms is another like term whose co-efficient is the sum of
coefficients of the given terms
This rule of addition is also applicable to more than two like terms
To add 3mn2,
5mn2, 4mn2, we have
3mn2 + ( 5) mn2 + 4mn2
= [3 + ( 5) + 4] mn2
= [7 + ( 5)] mn2
= 2mn2
Let us subtract 7x from 9x.
9x – 7x = (9 7)x
= 2x
Similarly, 11a – 3a = (11 – 3) a
= 8a
3a – 11a = (3 – 11) a
= 8a
9pq – 3pq = (9–3)pq
= 6pq
( 2pq) – 3pq = ( 2 3) pq
= 5pq
16m2n
11nm2 = (16
11)m2n
Thus,
The difference of two like terms is another like term whose coefficient is the
difference of the coefficients of the given like terms.
13
Addition and Subtraction of Unlike Terms
Recall that 3x and 7 are unlike terms.
If we add them, we write 3x+7.
Similarly, if we subtract 7 from 3x, we write
3x
7, or if we subtract 3x from 7, we write it as 7
3x.
Addition and subtraction of Algebraic Expressions
Addition
We explain addition of algebraic expressions through some examples:
(i)
Add 2x + 9 and 5x – 3
(2x + 9) + (5x – 3) = 2x + 9 + 5x – 3
= (2x + 5x) + (9 3)
[Combining like terms]
= (2+5)x + 6
[Constants 6 and ( 3) are
= 7x + 6
considered as like terms]
Thus, (2x + 9) + (5x–3) = 7x + 6
Alternatively, we can also do it as follows:
2x + 9
+ 5x – 3
[Writing the expressions in a way that like terms
are in the same column]
(2+5)x + (9 3)
or
(ii)
7x + 6
Add 4x + 3y – z and 5y – 2x
We have
(4x + 3y – z) + (5y – 2x)
= 4x + 3y – z + 5y – 2x
= (4x 2x) + (3y + 5y) –z
[Combining like terms]
= (4 2)x + (3+5)y – z
= 2x + 8y – z
14
Thus,
(4x + 3y – z) + (5y – 2x) = 2x + 8y – z
Alternatively,
4x + 3y - z
[Arranging like terms in the same column. There is
2x + 5y
no like term corresponding to the term
z
2x + 8y - z
In the above examples, the first method is known as horizontal method and the other is
known as vertical or column method.
(iii)
Add a2 – b2 – c2,
c2 – a2
b2 and b2 – c2 – a2
We write these expressions in column as follows:
a2 – b2 – c2
[Writing like terms in the same column]
a2 – b2 + c2
a2 + b2 – c2
a2 – b2 - c2
[a2 – a2 – a2 = -a2] and so on.
Thus, (a2 – b2 – c2) + (c2 – a2 – b2) + (b2 – c2 – a2) = a2 – b2 – c2
Subtraction
We explain subtraction of algebraic expressions through some examples:
(i)
Subtract 5x – 3 from 2x + 9
We have
(2x + 9) – (5x 3) = 2x + 9 – 5x + 3
= (2x
[Observe that sign of each term of the
expression (5x – 3) has been changed
from ‘+’ to ‘ ‘ and vice versa]
5x) + (9 + 3) [ Combining like terms]
= (2 5)x + 12
= 3x + 12
15
Alternatively
2x + 9
[Note that sign of each term of expression to be
5x – 3
–
+
3x + 12
subtracted has been changed]
[Adding 2x and ( 5x) and 9 and (+3)
(ii) Subtract 2x + 9 from 5x – 3
we have
5x - 3
(5x – 3) – (2x + 9) = 5x – 3 – 2x – 9
= (5x – 2x) – (3 + 9)
2x
9
3x
12
= 3x – 12
(iii)
Subtract –x2 – 2xy + y2 from 3x2 – 4xy – 5y2
3x2 – 4xy – 5y2
x2
2xy + y2
[sign of each term changed]
+ +
4x2 – 2xy – 6y2
[3x2 + x2 = 4x2, 4xy + 2xy = 2xy and 5y2 – y2 = 6y2]
Thus,
To subtract an algebraic expression from a given expression, we change sign
of each term (from ‘+’ to ‘ ‘ and ‘ ‘ to ‘+’) of the expression to be subtracted
and then add the two expressions.
Example 6: Add x2 + x + 1, 2x + 5,
9
4x2
Solution:
x2 + x + 1
+ 2x + 5
4x2
3x2 + 3x
9
3
16
Example 7: Subtract – m2 + 8 from 5m3 – 3m – 2m2
Solution:
5m3 – 2m2 – 3m
m2
8
[Changing sign of each term]
5 m3– m2 – 3m – 8
Example 8: Subtract the sum of 3x + 2y + z and 5x – 3y + 7z from the sum of
and 2y – 3z.
4x + 3y
Solution: Let us first find the sum of 3x + 2y + z and 5x – 3y + 7z and the sum of
4x + 3y and 2y 3z.
4x + 3y
3x + 2y + z
+ 5x – 3y + 7z
2y – 3z
8 x – y + 8z
4x + y – 3z
Now, we have to find ( 4x + y – 3z) – (8x y + 8z)
we have:
4x + y
– 3z
8x
y + 8z
+
12x + 2y – 11z
Multiplication of Algebraic Expressions
Multiplying a monomial by a monomial
Consider the monomials 4x, 9
we have (4x) x ( 9) = (4 x) ( 9)
= 4 (–9) x
= ( 36) x
=
Similarly, x
5y
[x is also a number so, we
can use commutative and
associative properties]
36 x
=x
5
y
=5xxxy
= 5 x xy
= 5xy
and (3x) x ( 4y)
= (3xx) x ( 4) x y
17
= [3 x ( 4)] x x x y
= ( 12) x xy
= 12xy
( 9x) x (5 x3)
(3a) x ( 4abc)
= ( 9) x x x 5 x x3
=
[(9) x 5] x [x x x3]
=
45 x (x3+1)
=
45 x x4
=
45 x4
[using law of exponent xm x xn = xm+n]
= (3xa) x ( 4) x a x b x c
= [3 x (- 4)] x a x a x b x c
= ( 12) x a2 x b x c
= 12 a2 bc
From the above examples, note that
(i)
Coefficient of the product = Product of the coefficients of two monomials
(ii)
Algebraic (literal) part of the product = Product of the algebraic parts of
two monomials
This product rule can be extended to find the product of more than two monomials.
For example,
(ab2) x (3ab) x ( 9a4b4)
= [(1) x (3) x [( 9)]
x
[(ab2) x (ab) x (a4 b4)]
Product of coefficients
x
Product of algebraic parts
= ( 27) x [(a x a x a4) x (b2 x b x b4)]
= ( 27) x (a1+1+4) x (b2+1+4)
= ( 27) x a6 x b7
=
27 a6 b7
18
Multiplying a monomial by a binomial
We explain it through examples.
(i)
Consider the monomial 5x and the binomial 3x – 2
We have
(5x) x (3x 2) = [5x x 3x] – [5x x (2)]
= [(5 x 3 x x x x)]
Recall 5x, 3x 2 are numbers
[(5) x (2) x x]
So, we can use distributive property
= 15x2 -10x
(ii)
Consider the monomial x2y and binomial 2xy – 7x3 y4
We have
(x2 y) x (2xy – 7x3 y4)
= [(x2y) x (2xy)] – [(x2y) x (7x3y4)]
[Using distributive property]
= [2 x (x2 x x) x (y x y)] – [7 x (x2 x x3) (y x y4)]
= 2 x3 y2 – 7x5 y5
Example 9: Find the product of
(i)
8 y2 z and 4 xz2
(ii)
7 m3 n2 and 12 m + 11n2 m2 p
Solution:
(i)
( 8y2z) x (4xz2) = ( 8) x (4) x (y2z) x (xz2)
=
(ii)
32 xy2z3
( 7m3n2) x ( 12m + 11n2 m2p)
= ( 7m3n2) x ( 12m) + ( 7m3n2) x (11n2m2p)
= ( 7) x ( 12)m4 n2 – 77 m5 n4 p
= 84 m4n2 – 77 m5 n4 p
Multiplying a binomial by a binomial
For multiplying a binomial by a binomial, we multiply each term of the first
binomial using distributive property two times.
19
We explain it through some examples.
(i)
(3x + 2y) x (2x + 3y)
= (3x) x (2x+ 3y) + 2y (2x + 3y)
= (3x) x (2x) + (3x) x (3y) + (2y) x (2x) + (2y) x (3y)
= 6x2 + 9xy + 4xy + 6y2
= 6x2 + (9 + 4) xy + 6y2
= 6x2 + 13xy + 6y2
(ii)
(2x
5) x (2x + 1)
= 2x (2x + 1) – 5 (2x+1)
= (2x x 2x) + (2x) x (1) – (5 x 2x) – (5) x (1)
= 4x2 + 2x – 10x 5
= 4x2 – 8x 5
5. Algebraic Expression, Equation, Formula and Identity
Algebraic Expression
You are already familiar with algebraic expressions and how they are
formed.
For example 3x + y is an algebraic expression.
Similarly 2 + 2b, 3x – 2, x3 – yz, + 3xyz etc. are all algebraic expressions.
Equation
If we equate two algebraic expressions using the sign of equality (=), then
what we obtain is called an equation.
3x – 2
=
algebraic expression
5x – 6
algebraic expression
For example 3x – 2 = 5x – 6,
p = 2 + 2b , A =
xb
3x + y = 1, y – 2x = x + y
a2 + b2 = b2 + a2
20
(x+1) x (x+1) = x2 + 2x + 1, a(a b) = a2 – ab, etc., are all equations
Identity
Of the above equations,
3x – 2 = 5x – 6
is true for x = 2 which means if we put the value of x as 2 in LHS and RHS,
then LHS = RHS. [LHS = 3x – 2 = 3x2-2 = 6-2 = 4
RHS = 5x-6 = 5 x 2– 6 = 10-6 = 4]
But if we put x = 1, then LHS = 3 x 1 = 1
RHS = 5 x 1
Thus LHS
6= 1
RHS
Recall that x = 2 is a solution of this equation
Now consider the equation
x x (x+1) = x + x2
Check that if you substitute any value of x in this equation then LHS = RHS
But this was not true in the previous case i.e.,
3x–2 = 5x – 6
The equations like x x (x+1) = x + x2
which are true for all the values of the variable are called identities.
a(a-b) + ab = a2
is also an identity (Why?)
Formula
Of the above equations, look at the equations :
p = 2 + 2b
You will find that it is true when =5, b=3, p=16.
It is also true for
= 8, b = 4, p = 24,
but it is not true for p = 25,
= 8, b = 4
21
Thus, p = 2 + 2b is not an identity.
If you carefully look at this equation, you will find that in LHS of this
equation is related to and b (on RHS) in the sense that
Perimeter of a rectangle = 2 x length + 2 x breadth.
Such an equation is called a formula.
This is, clearly, a formula for perimeter of a rectangle.
Similarly A =
xb
is also a formula. It gives area of a rectangle in terms of its length and
breadth.
Clearly p = 3s
is a formula for perimeter of an equilateral triangle with side s.
Not that some equations are not true for all the value (s) of the variable (s).
Usually these type of equations are referred to as conditional equations.
The word equation is generally used only for conditional equations and
they are true for some specific value (s) of the variable (s)
Example 10: Identify which of the following are expressions, equations, formula or
identities:
(i)
3x – x+2 = 5 3 + 2x
(ii)
p = 4a
(iii)
(x+y) + 3x
(iv)
(x-1) x (x-1) = x2 + 1 2x
(v)
9y + 1 = 6y
(vi)
A=
(vii)
I=
(viii) 3x – x – 2 = 5
(ix)
3 + 3x
5x x (2x 3)
22
Solution:
Algebraic expression are (iii) and (ix)
All are equations except (iii) and (ix)
Formula are:
(ii), (vi) and (vii), (ii) is for perimeter of a square, (vi) is for area of a triangle and (vii)
is for simple interest.
Identities are (i) and (iv).
Note that (ii), (v) , (vi), (vii) and (viii) are conditional equations.
6. Linear Equations
Linear Equations in one variable
You have already been introduced to linear equations in one variable. Following are
some examples of linear equations.
(i)
x + 5 = 10
(ii)
12 = 7 + y
(iii)
=5
(iv)
2m = 12
(v)
3x = 9 – 2x
From the above, you can observe that the variable involved in each of the above
equations is one. Further the exponent of the variable is also one. Only such
equations are called linear equations in one variable. Thus, an equation in one
variable with only 1 as the exponent of the variable is called a linear equation is
one variable.
For example, 5x 4 = 11, 12t–5 = t etc., are linear equations in one variable. But the
equations such as
3x – 2y = 9, and x2 = 25,
are not linear equations in one variable.
3x – 2y = 9 is a linear equation in two variables x and y (as the exponents of x and y is
only 1).
23
x2 = 25 is an equation in one variable, but it is not linear, because the exponent of the
variable x is 2, not 1.
Solving a linear Equation
Recall that an equation may be compared
with a balance.
Its sides are two pans and the equality sign
(=) tells us that the two pans are in
equilibrium.
If we put (or remove) equal weights from
both the pans, then the balance remains
undisturbed. Similarly, in the case of an
equations, it will not change if we
(i)
add (or subtract) the same number to (or from) both sides of the
equation.
(ii)
Multiply (repeated addition) [or divide (repeated subtraction)] both
sides of the equation by the same non-zero number.
We explain the process of solving an equation through some examples:
Example 11: Solve the following equations:
(i)
x + 5 = 10
(ii)
12 = 7 + y
(iii)
=x 5
(iv)
3x = 9 2x
(v)
2 (y+1) = 10y – 14
Solution:
(i)
x + 5 = 10
or
x + 5 – 5 = 10-5 (Subtract 5 from both sides)
or
x + 0 = 10
or
x + 0 = 10 5
or
x=5
5
(A)
24
So, x = 5 is the solution of the equation
Verification : LHS = x + 5 = 5 + 5 = 10
RHS = 10
So, LHS = RHS
Hence, verified
(ii)
12 = 7 + y
or
12 7 = 7+y 7 (Subtract 7 from both sides)
or
12 7 = y
(B)
5=y
So, y = 5 is the solutions of the equation
Verification : LHS = 12
RHS = 7+y = 7+5 = 12
So, LHS = RHS
Hence, verified.
Note: From (A) of (i), you can observe that term 5 from LHS of the equation has been
shifted to RHS with sign changed (-5).
Similarly, from (B) of (ii), term ‘7’ from RHS has been shifted to LHS in the form -7. In
both the cases, a term has been shifted from one side of the equation to its other side by
changing its sign. This process is called transposition of the term.
(iii)
=x
5
or
x 4 = (x
or
x = 4x 20
or
x + ( x) = 4x – 20 + ( x)
or
0 = 3x -20
or
0 + 20 = 3x
or
20 = 3x
5) x 4
20 + 20
(multiplying both sides by 4)
(adding (-x) on both sides)
(Adding 20 to both sides)
25
or
=x
(Dividing both sides by 3)
Verification: LHS =
=
=
RHS = x 5
=
5=
–
=
So, LHS = RHS
Hence, Verified
(iv)
3x = 9 2x
or
3x + 2x = 9[transposing ‘-2x’ from RHS to LHS]
or
5x = 9
or
x=
Verification: LHS = 3x = 3 x =
RHS = 9 – 2x = 9
2x =9
=
=
So, LHS = RHS
Hence, Verified
(v)
2(y+1) = 10y – 14
or
2y + 2 = 10y – 14
or
2y + 2 – 10y = –14 [transposing ‘10y’ from RHS to LHS]
or
– 8y + 2 = 14
or
-8y = 14 2 [transposing ‘+2’ from LHS to RHS]
or
-8y = 16
or
y=
Verification:
=2
LHS = 2 (y + 1)
26
= 2 (2+1)
=6
RHS = 10y – 14
= 10 x 2 14 = 6
So, LHS = RHS
Hence, Verified
7.
Applications of Linear Equations
You have already learnt how to solve a linear equations in one variable. Now, you
will learn to solve problems from practical situations. We explain it through some
examples.
Examples 12: The sum of two numbers is 97. One number is 19 greater than the
other numbers. What are the two numbers?
Solution: Our first step to solve such question is to identify the quantity or
quantities you are being asked to find. Here, we have to find the two numbers.
• If we let x stand for the smaller number, the larger number will be x+19.
So,
x+(x+19) = 97
[Sum of the two numbers is given as 97]
• Solve this equation
x+x+19 = 97
or
2x+19 = 97
or
2x = 97-19
or
2x = 78
[Transposing 19 from LHS to RHS]
x = 39
Thus, the smaller numbers is 39 and the larger number is 39+19=58
27
Check:
Smaller number =19
Larger number = 58
Sum = 19+58 = 97
Example 13: The sum of two numbers is 47. If one number is 2 more than 4 times
the other, find the numbers.
Solution: Let the smaller number be x
Then, the larger number
= 4 times x + 2
= 4x + 2
The sum of two numbers
So,
= x + (4x+2)
x + (4x+2) = 47
[It is given that the sum = 47]
Now, we have to solve this equation.
x+4x+2 = 47
or
or
5x+2 = 47
5x = 47–2
[transposing 2 to RHS]
5x = 45
or
x=9
thus, the smaller number is 9, and
the larger number is 4x+2
=4x9+2
= 38
Check: sum = 38 + 9 = 47
Example 14: A farmer wishes to put up a fence around his rectangular field. The
length of the field is 150 meters more than its width. If the farmer has 1900 meters
of fencing, what are the dimensions of the field?
Solution: Let x be the width of the field
Then, the length of field = x+150
28
Perimeter of field = 2 +2w
= 2(x+150)+2x
Length of fencing= 1900 meters
So,
2x+2(x+150) = 1900
or,
2x+2x+300 = 1900
or,
4x + 300 = 1900
or,
4x = 1900–300
or,
4x = 1600
[transposing 300 to RHS]
or,
=
[Dividing both sides by 4]
or,
x = 400
Thus, the width of the field = 400 m
and the length of the field = 400 + 150 = 550 m
Check: Perimeter of field
= [2 x (400) + 2 x (550)]m
= (800 + 1100) m
= 1900 m
Example 15: Anvi is three years older than Hasan. Six year ago, Anvi’s age was 4
times Hasan’a age. Find their ages.
Solution: Let age of Hasan be x year now
Then the age of Anvi is three years more i.e. x+3
6 years ago
So,
Hasan age
=
x–6
Anvi’s age
=
(x+3) – 6
=
x+3–6
=
x–3
=
4(x–6)
x–3
29
[Anvi’s age is 4 times Hasan’s age]
Let us solve this equation.
x–3 = 4(x–6)
or
x–3 = 4x–24
or
x–3–4x = 24
or
x–4x–3 = –24
or
–3x = –24+3
or
–3x = –21
or
=
[transposing 4x to LHS]
[transposing –3 to RHS]
[Dividing both sides by –3]
So, Hasan’s age = 7 years and Anvi’s age = (7+3) yeas
= 10 years
Check: Anvis age 10 years is 3 year more than Hasan’s age i.e. 7 years
6 year ago :
Anvi’s age = (10–6) = 4 years
Hasan’s age = (7–6) = 1 years
So, Anvi’s age = 4 times Hasan’s age.
Example 16: A person sells a cap for ì 60 and a pair of socks for ì 45. On a
particular day, his total sale was ì 960 and the number of items he sold was 18.
Find the number of caps and number of pairs of socks.
Solution: Let the number of caps sold be x
Then, the number of pairs of socks = 18–x
Sale from caps = ì 60 x x = 60x
Sale from socks = ì 45 (18–x)
According to question,
60x+45(18–x) = 960
or
[Total sale = ì 960]
60x+45 x 18–45x = 960
30
or
60x–45x + 810 = 960
or
15x = 960 – 810
or
x=
or
x = 10
[transposing 810 from LHS to RHS]
So, the number of caps = 10 and the number of pairs of socks = 18–10 = 8.
Check:
Sale from caps = ì 10 x 60 = ì 600
Sale from socks = ì 8 x45 = ì 360
Total sales
= ì 600 + ì 360
= ì 960
31
STUDENT’S
SUPPORT
MATERIAL
32
STUDENT’S WORKSHEET - 1
Variable and Constants
Warm up W1
Name of the student ______________________
Date ______
Activity – Basket of Expressions
Tom and Tim saw a tank full of variables and constants.
Tom is to collect the variables and Tim is to collect constants
from the tank.
The one who does it first is the winner.
3
-3x
3x+2y
5+2
xy 4x
5z
52
2xyz
xu3+2+ -g
2y
5
2x+3x
Write here what they have collected.
Tom’s collection
Tim’s collection
33
STUDENT’S WORKSHEET - 2
Forming Expressions
Warm up W2
Name of the student ______________________
Date ______
Activity – Calendar Expression
DECEMBER
M
T
W
T
F
S
S
7
14
21
28
1
8
15
22
29
2
9
16
23
30
3
10
17
24
31
4
11
18
25
5
12
19
26
6
13
20
27
1. In the given calendar some of the dates are highlighted. Taking 17as the ‘BASE
DATE’ i.e. x, write expression for all the dates which are highlighted here. The
first one is done for you.
Date
9th Dec
Expression
x-8
2. Choose any 9 dates that form a square, like the one highlighted.
a) Add all nine numbers and then divide by the number in the middle.
What do you get?
34
b) Repeat with another set of 9 dates that form a square. What do you
observe?
c) Let’s ask Ms. Expression for an explanation.
Ms. Expression: Use the previous table to complete the grid, take the
number in the center as n. Then the nine numbers would be
Ms. Expression: Firstly, let's simplify the expression by adding like terms and
then divide by the number in the middle, which in this case is n. Go ahead
and solve the mystery.
3. Samantha has joined weekly dance classes. If her first class for the month is on 6th of
December i.e. Sunday, write her other dance classes using expressions, by taking
6th of December as y.
35
4. John always plans his Christmas weekend in advance. Help him plan his schedule,
taking 25th December as ‘C’.
36
STUDENT’S WORKSHEET - 3
Types of Algebraic Expressions
Pre CONTENT WORKSHEET P 1
Name of the student ______________________
Date ______
Activity – Funnel Act
From the funnel below, filter out the algebraic expressions as per their type:
Monomial
Binomial
Polynomial
37
STUDENT’S WORKSHEET - 4
Degree, Terms and Coefficients
Pre CONTENT WORKSHEET P 2
Name of the student ______________________
Date ______
Activity – Express the Expression
A)
Write as monomials, binomials or trinomials for the expressions
given below
Expression
Monomial/bin Degree of the
omial/trinomi polynomial
al/polynomial
4
2x+5y x2
3xyz
2x2+3x
-2y+3yz+4x
0
9y+3x2
x-2y+4
Z3+ 2
3xz3+4x3
38
Variable(s)
Constant
term( if
any)
B)
Complete the given table by writing the terms and their factors for the given
algebraic expressions. One is done for you.
C)
In the given algebraic expressions write the required co-efficient:
1.
-3x+b+5-4x2
a)
of x
b)
constant term
c)
of x2
39
2.
3.
4
5.
6.
b + x2
a)
of b
b)
of x
-3ab-6ax2-4b2
a)
of x
b)
of a
c)
of b
-xy-4xy2
a)
of xy
b)
of y2
4n2+mn-4 m3
a)
of mn
b)
of n
c)
of m3
a2-b3
a)
of b
b)
of a2
40
STUDENT’S WORKSHEET -5
Algebraic expressions through Pattern
CONTENT WORKSHEET C1
Name of the student ______________________
Date ______
Activity- Pattern Express
A. Forming ‘letter P’ using the matchsticks.
Draw 1, 2, 3 ‘attached P’ in the space provided.
(i)
(ii)
(iii)
1.
How many matchsticks did you use in fig (i)? _______
2.
How many more matchsticks than fig (i), did you use in fig (ii)? _______
3.
Complete the following table showing the number of matchsticks required to form
the sequence of letter P
No. of letters ‘ P’ = n
No. of matchsticks = m
1
2
3
5
8
10
4. In words, write the relationship between the number of matchsticks (m) and the
number of letters ‘P’.
______________________________________________________________________
5. How many matchsticks do you need to make a sequence of 30 P’s?
______________________________________________________________________
6. How many P’s can you form with 36 matchsticks?
______________________________________________________________________
41
B. Form the following sequence of squares using matchsticks.
Fig 1
Fig 2
Fig 3
1. Complete the following table showing the number of matchsticks required to
build the square sequence.
No. of squares = s
No. of matchsticks = m
1
2
3
5
2. In Fig 2, how many more matchsticks are required than in Fig 1?
________________________________________________________________
3. Similarly in Fig 3, how many more matchsticks are required than in Fig 2?
________________________________________________________________
4. With every new addition of a square in the sequence, by what multiple is the
number of matchsticks required?
________________________________________________________________
5. Can we say, For 1 square we need (3 × 1 + 1) matchsticks.
For 2 squares we need (3 × 2 + 1) matchsticks.
6. In words, write the relationship between number of squares built (s) and the
number of matchsticks used (m).
7. Complete the algebraic expression using the variables to represent the above
pattern:
42
No. of matchsticks = _______ times the no. of squares + ________
8. How many matchsticks are required to build 100 squares?
________________________________________________________________
9. How many matchsticks are required to build ‘s’ squares?
________________________________________________________________
C.
Investigate a repeating pattern:
Fig (i)
fig (ii)
Answer the following questions:
1. How many toothpicks did you use in fig (i)?
2. Complete the following table showing the number of toothpicks to build the
pattern.
3. Describe the relationship between the number of toothpicks (t) and the number
of repeating units (n).
4. Use algebra to write this relationship rule.
5. How many repeated units could you build with 100 toothpicks?
43
STUDENT’S WORKSHEET -6
Like and Unlike Terms
CONTENT WORKSHEET C2
Name of the student ______________________
Date ______
Activity- No Debate
I have two cards with 3x
and -4xy.
I have 5x and -5yx cards.
Take my card having 5yx and give me 3x, so
that we get like cards.
A) Choose and write like terms together.
3xy
-5x
-3xy
-4
67
-4xy
3
6x
-4x
B) Make 8 more cards like the sample card i.e. using like terms
-5xy
3y
-y
13xyz
44
-5y
7y
x
C) In the given space add the above created like terms (including the sample card) by
showing complete working.
45
STUDENT’S WORKSHEET –7
Addition of Algebraic Expressions
CONTENT Worksheet C3
Name of the student ______________________
Date ______
Activity – Dancing Dice
http://www.bgfl.org/bgfl/custom/resources_ftp/client_ftp/ks1/maths/dice/
Go to the given link and create your own virtual dice by selecting a 6-sided or 8-sided dice.
Dice of Monomials
Task 1: Dancing Select option of 6-sided or 8-sided dice.
Create your own dice by entering some monomial expressions of your choice
and press ‘create’ (as shown below).
Start rolling your dice… roll 5 times and record your outcomes.
Add your 5 outcomes using like terms and record your working in the given
space.
Create one more dice using monomials and repeat the task.
Dice 1:
Dice 2:
46
Task 2: Dancing Dice of Binomials
Select option of 6-sided or 8-sided dice.
Create your own dice by entering some binomial expressions in more than one
variable, of your choice and ‘create’ your own dice(as shown below).
Start rolling your dice… roll 3 times and add your outcomes vertically or
horizontally in the space provided.
Repeat it once more.
Sum 1:
Sum 2:
47
STUDENT’S WORKSHEET –8
Subtraction of Algebraic Expressions
CONTENT Worksheet C4
Activity – Domino Activity
Cut, begin with the 'Start' domino and then add dominoes so that the matching
expressions are equivalent to form a continuous loop.
Contd.
48
49
STUDENT’S WORKSHEET –9
Skill Drill
CONTENT Worksheet C5
Name of the student ______________________
Date ______
Activity -Test Mania
Simiplify the following expression (1 to 3)
1.
2.
3.
4. Subtract
5. Subtract
6. Simplify:
7. Simplify:
50
8. Simplify:
9. Simplify:
from
10. Simplify:
11. What should be added to
to get
12. What should be added to
to get
13. What should be subtracted from
14. Subtract the sum of
?
?
to get
and
51
?
from
.
STUDENT’S WORKSHEET –10
Addition and Subtraction of Algebraic Expressions
CONTENT Worksheet C6
Name of the student ______________________
Date ______
Activity - Algebra cards
Materials Required:
Prepare the algebra cards having variables x, y, z, x y, y z, z x as:
x, 2x, 3x, 4x, … ,13x
y, 2y, 3y, 4y, … , 13y etc.
Number of players: 2 to 4 players
Sample:
x
y
zy
xy
zx
2x
2y
2zy
2xy
2xz
How to play:
1.
Shuffle the algebra cards, and then place them on the table, face down.
2.
Take turns to pick up a card and ‘open’ it.
3.
After a minimum of 4 cards are ‘opened’, players have to observe if using the
‘opened’ cards and some arithmetic operators ‘+’, ‘-’ etc., they can frame an
addition or subtraction problem, having an answer as some open cards.
Player will tabulate their working and collect the used cards. For e.g.:
A player may use 2x, 3x, 4y, 3y, 5x, y; as given below and collect them
(‘open’ cards are highlighted with a different color).
(2x + 4y) + (3x – 3y) = 5x + y
11 x
3xy
y
4z x
52
3x
2x
2y
2zy
2xy
4y
5x
3y
5.
If a player is unable to frame the problem correctly, he or she cannot collect
any card.
6.
If the players ‘open’ all of the cards in the deck and no further problem is
possible, they stop the game.
7.
The player with maximum number of cards wins.
Tabulate your play progress:
PLAYER 1: Working
Cards picked
(2x + 4y) + (3x – 3y) = 5x + y
2x, 3x, 4y, 3y, 5x, y
PLAYER 2: Working
Cards picked
Make as per number of players.
53
STUDENT’S WORKSHEET –11
Multiplying monomial with monomial
CONTENT Worksheet C7
Name of the student ______________________
Date ______
Activity 1 - Pyramid pointers
Which expression goes on top point of pyramid C?
Show working-
54
Activity 2– Lines of Thought
Put your thinking caps on and workout which expression is missing from each line
of hexagons. Use the first line as an example
55
STUDENT’S WORKSHEET –12
Multiplying a monomial with a binomial
CONTENT Worksheet C8
Name of the student ______________________
Date ______
Activity 1-Solve the standard triangular jigsaw puzzle.
56
Activity 2-Multiply the monomial with the binomial:
-3w ( -8x + z ) =
.
( 6m – 9n ) . ( -m2 ) =
7xy .( x2 – y3 ) =
.
.
57
STUDENT’S WORKSHEET –13
Multiplication of Binomial Algebraic Expressions
CONTENT Worksheet C 9
Name of the student ______________________
Date ______
Activity - Area Arena
Apply distributive property to multiply algebraic expressions and find area of given shapes.
1.
2.
4x-3
2 x- y
3.
4 x -3
2x-3
4.
5a-3b+2
5.
6.
13-3m
2a+b
5x-y
2y+x
2+m
3x-5y
7.
8.
5x–3y+z
2x–5
Space for your working:
58
STUDENT’S WORKSHEET – 14
Evaluate Algebraic Expressions
CONTENT Worksheet C10
Name of the student ______________________
Date ______
Activity - Operation Expression
Mr. X is here to investigate the given expression for the different
values of variable.
Help him evaluate algebraic expression by substituting the given
values.
2
3y – 4 y + 5
1. Mr. X : I suspect that y= -4. What would be the value of the Expression?
2. Mr. X : I suspect that y= -1. What would be the value of the Expression?
3. Mr. X : I suspect that y= 5. What would be the value of the Expression?
4. Mr. X : I suspect that y= 3. What would be the value of the Expression?
59
5. Mr. X : I suspect that y= -2. What would be the value of the Expression?
6. Mr. X : Help me evaluate 3x2-5x+2, for x= -1.
7. Mr. X : I have to solve a few more cases where a=-2, b=3 and c=-1
a)
a2 + 4b2- 6b
b)
3a2 - b2- ab
c)
5-2b2-4a
d)
4 - a2 + b2- b
e)
5a2 -2b2- 6ab
60
STUDENT’S WORKSHEET –15
Expression, equation, formula and identity
CONTENT Worksheet C11
Name of the student ______________________
Date ______
Activity 1- Distinguishing game
From the expressions given in the basket separate the algebraic
expression,equation,formula and identity and write them in the columns provided.
Algebraic expression
Identity
Equation
61
Formula
What do you observe?
……………………………………………………………………………………………………
……………………………………………………………………………………………………
What is an algebraic expression?
……………………………………………………………………………………………………
……………………………………………………………………………………………………
What is an identity?
……………………………………………………………………………………………………
……………………………………………………………………………………………………
What is an equation?
……………………………………………………………………………………………………
……………………………………………………………………………………………………
62
What is a formula?
……………………………………………………………………………………………………
……………………………………………………………………………………………………
Therefore, the expressions of the type x+y=5 and
are called
___________________ because they are true for only some sets of values of x and y.
The expression 2a-b and 4x-2+3 are called _________________________ because
constants and variables are joined by mathematical operations.
The
expressions
of
the
type
(a+b)2
_____________________________because they are
a and b.
=a2+b2+2ab
are
called
true for all values of
The expressions of the type area of square=side2 is called
___________________because it typically describes a calculation.
63
STUDENT’S WORKSHEET –16
Simple Linear Equations
CONTENT Worksheet C12
Name of the student ______________________
Date ______
Activity 1- Formulate equations
STATEMENT
EQUATION
The product of 4 and x is 24
The sum of 4y and 5 is 25
7 taken away from x is 9
One-fifth of a number plus 7 is 30
Thrice of a number increased by 8 gives 26
Product of 5 and x is 35
Twice a number diminished by 7 is 13
64
Now answer the following:
1. Name the variables used above.
……………………………………………………………………………………
2. How many variable(s) are involved in each expression?
……………………………………………………………………………………
3. What is highest power of the variables involved?
……………………………………………………………………………………
4. Does each expression contain an equality sign?
……………………………………………………………………………………
Thus an algebraic expression with………variable having highest power as
………………. and an ………………….sign is called a simple equation.
Activity 2- Form more equations
Consider a weighing balance as shown.On the left side there is a triangle and some
beads.On the right side other beads, the weight of both sides is same. Form an equation
to represent this equality by taking triangle as ‘x’
a)
Equation: …………………………………………………………………
65
b)
Equation: …………………………………………………………………
c)
Equation: …………………………………………………………………
d)
Equation: …………………………………………………………………
66
Activity 3- Convert the following equations in statement form:
Equation
Statement
18 + x = 40
x/6 = 12
2x-7=15
3y=21
x+5=10
3a+2=5a
2/3.x=14
33-x=11
67
STUDENT’S WORKSHEET –17
Solving Simple Linear Equations
CONTENT Worksheet C13
Name of the student ______________________
Date ______
Activity 1- Solving simple equations:
Whatever you do to an equation,
do the S A M E thing
to B O T H sides of that equation!
For eg.
x+3=6
-3
-3
x+0=3
Now solve the following in the same manner:
i)
x – 4 = 13
ii)
y + 12 = 3
iii)
3x – 13 = 2
iv)
3 = 15
68
v)
b/2 = 41
vi)
2s + 6 = 12
vii)
3p + 12 = 0
viii)
2a/3 – 3 = 5
Activity 2- Transposition Method of solving an equation .
For eg. 2x + 6 = 7
Note: Transpose the term containing no variable on the other side.In doing so its sign
changes
2x = 7 – 6
2x = 1
x=½
Now solve the following and check your answer by subsitution:
i)
6x + 10 = -2
check
69
ii)
3a/2 = 2/3
check
iii)
12y – 5 = 25
check
iv)
2m – ½ = -1/3
check
v)
x/5 + 3 = 1
check
STUDENT’S WORKSHEET –18
Word problems
CONTENT Worksheet C14
Name of the student ______________________
Date ______
Activity 1- Solving simple equations:
70
1.
Sam subtracts thrice the number of notebooks he has from 50.He finds them to be
8.
2.
Ramesh scored twice as many runs as Rahul. Together their score were two short
of a double century. How many runs did each one score?
3.
Reena’s father is 49 years old. He is 4 years older than three times Reena’s age.
What is Reena’s age?
71
4.
Amit says he has 7 marbles more than five times the marbles Mohan has. Amit has
37 marbles. How many marbles Mohan has?
5.
After buying a book, Rajat is left with rupees 5 out of a fifty-rupee note. What is
the cost of the book?
6.
Meena thinks of a number. If 60 is subtracted from it the result is 4. Find the
number.
7.
When you divide a certain number by 13, the quotient is -18 and the remainder is
7. Find the number.
72
8.
If one side of a square is represented by 4y-7 and the adjacent side is represented
by 3y+5, find the value of y.
9.
Each of the two equal sides of an isosceles triangle is three times as large as the
third side. If the perimeter of the triangle is 28 cm, find each side of the triangle.
73
STUDENT’S WORKSHEET –19
Simple Equations with variables on both sides
CONTENT Worksheet C15
Name of the student ______________________
Date ______
Activity 1- More simple equations
Mandi has to solve the equation written on the roof of house so as to locate the door and
enter.Help her to get inside.(Do the working in the square)
i)
ii)
74
iii)
iv)
v)
75
vi)
vii)
viii)
76
STUDENT’S WORKSHEET – 20
Independent Practice
POST CONTENT Worksheet PC 1
Name of the student ______________________
Date ______
Activity - Test Your Knowledge
1.
Simplify 2xy-5-2y+yx-3y+6-yx-4y-3xy+2
2.
Subtract 4x2+2x-11 from 3x2-4x+8
3.
Let P= a2- b2 - 2b, Q=a2 + 4b2- 6b, R=b2 + 6, S=a2-4b and T=-2a2+b2 - b
Find
4.
a)
P+Q+R
b)
R+S–T
c)
Q–R+S
Find the value of x
6(1-4x) + 7(2+5x) = 53
5.
Verify that x=2 is a root of the equation 5x-12=-2
77
2y 11
3y 2
4
6.
Solve
7.
A number is two more than 3/5 of itself. Find the number.
8.
The sum of two consecutive odd numbers is 68. Find them.
9.
From the sum of 3x2-5x+2 and –5x2-8x+6, subtract 4x2-9x-7
10.
If x=1, y=-2 and z=3, find the value of
a)
x3+y3+z3-3xyz
b)
3xy4-15x2y+4z
11.
Add 5x2-7x+3, 8x2+2x-5 and 7x2-x-2
12.
If a=3, b=2 prove that a3-b3=(a-b)(a2+ab+b2)
13.
What will be the value of a if 3x2+x+a equals 8 when x=1?
14.
Subtract the sum of 4b2+3c2 and 2b2+bc-6c2 from the sum of b2-2bc and c2-2bc-b2
15.
Divide 184 into two parts such that one third of one part may exceed one seventh
of the other part by 8.
78
16.
A dealer earned a profit by 5% by selling a radio. If he earned Rs. 500 profit, find
the cost price of the radio.
17.
If
18.
After 16 years, Fatima will be three times as old as she is now. Find her present
age.
19.
A number consists of two digits whose sum is 8. If 36 is added to the number the
digits interchange their places. Find the number.
20.
How many litres of 35% sugar solution should be added to a 17% sugar solution to
obtain 72 litres of 25% sugar solution
x -1
x 1
7
, find x.
9
79
STUDENT’S WORKSHEET – 21
Test your progress
POST CONTENT Worksheet PC2
Name of the student ______________________
Q1.
Date ______
Write the numerical coefficients, like terms and constant terms in their
respective columns.
Expression
Numerical Coefficients
15x+5-3x+8
23a – 6
7ab +8a – ab -b
9u +2 + u – 0.5u
17y – 4
3x – 7 + 6x -5
80
Like Terms
Constant term
Q2.
Simplify each expression by combining the like terms.
i)
3ab -2ac +4ab
ii)
2a -4b +7ab -5a +2b
iii)
7x -9y -3xy - 2y -x +4xy
iv)
5r -3s -t +4t +3s
v)
6(2p -1) - (p+5)
vi)
4(p +3q) - (7 +4q)
vii)
3(xy -1) + 3(x-2y) -4xy
viii) 7rs -2s -3(rs +1) -2s
81
ix)
Q3.
Q4.
9y -2z -(y-z)+3yz
Find the sum of the following expressions
i)
–x2 – 6xy , -x + 9xy , 3x – x2
ii)
9x5 +4x4 +20x3, -5x5 + 3x4 – 4x3
iii)
3x + 4y + 5z , 6x -5y -4z , -3x + 2y + 3z
iv)
-14u -17v , -10u – 18v
v)
-8m2n +3mn2 –mn , 6m2n +2mn2 – 2mn
Subtract :
i)
2xy from 6xy
ii)
-7a from 3a
iii)
6x +4y from 2x +3y
iv)
-9x -7y from -3x – 4y
v)
18m +12n from 12m – 30n
82
Q5.
Q6.
Q7.
Simplify the following expressions and find their values for given values of x
and y.
i)
3x2 –7x +3 – 4x , (for x = 2)
ii)
(x – y) +7x , (for x = 40 and y = 20)
iii)
-2(-3y + 2x) + y + (-x) (for x = -2 and y = 3)
iv)
6(2x) + 3(x – 4) , (for x = -1)
v)
(x – y) + (x + 9) , (for x = 18 and y = 8)
Find the product of following monomials
i)
-2xy , 15xy2z
ii)
-3a2 , 5ab3 and -b
iii)
9pqr2, -4pq2r and 7pr
iv)
5x2 , -10xy2
v)
x2 , 10xy3 ,
Simplify the following
i)
xy(x – y)
ii)
(a + b)7a2b2
iii)
3x3( 4x – 5y2)
iv)
x2 ( x2 – xy)
83
v)
Q8.
Q9.
–xy2(3xy2 – 5x2y)
Find the following products and simplify
i)
(x – y) (3x + 5y)
ii)
(a2 + b)(a + b2)
iii)
(
iv)
(p2 q2 – 2p2 )(5q2 – 7)
v)
(
-
)(
– 2) (
+
)
+ 2)
Rewrite each of the following phrases as an algebraic expression.
i)
The product of 12 and a number.
ii)
The quotient of 25 and ‘the difference of a number and 3’.
iii)
Sum of a number and its reciprocal.
iv)
Four times a number diminished by five.
v)
Fifteen less than product of seven and a number.
Q10. Subtract the sum of (8x -7y+ 5z2) and (3x – 4y –2z2) from the sum
of (2x +4y – 4z2) and (x – y – z2).
Q11. Find the value of the unknown.
a.
b.
84
c.
x 7 5
d.
e. x 6 58
Q12.
Solve the equation by inspection.
a.
b.
c.
d.
e.
Q13.
9 x 8
x 2 64 0
3x 21
x2
x
2
2
27
3 13
Complete the following equations by balancing on both sides.
a.
b. x 1
8
c.
Q14.
Determine by substitution if
a. -2 is the root of 5x=10
b. -5 is the root of x-5=0
c. 2 is the root of x3-8=0
Q15.
Solve the following.
a.
b.
c.
d.
e.
f.
9x=18
4a=12
-3y=-27
-18=-6x
-4v=8
-7x=3
85
Q16.
Frame the equations and solve.
a. One half times a number is 150. What is the number?
b. I am thinking of a number. Five times that number is 65. What is that
number?
Q17.
Solve.
a.
b.
c.
d.
e.
Q18.
Solve.
a.
b.
c.
Q19.
Simplify.
a. x 1 3
b. x 20
c. a 7
20
4
d. x 30
e.
y 20
f.
x 1 2
30
30
86
Q20. Find the root of each of the following equations.
a. 3x–2=5x–12
b. –22=14–9x
c. 9x+14=27
d. 18x=–13x+62
e. 13y=–12y+100
f. 12 y =7 y –15
g.
= 14
h. 30=6 8+y
i. 15(x–9)–2(x–12)+5(x+6) = 6
j.
–5=6
Q21.
Manoj is now 12 years old and Ram is 24 years old. How many years ago was
Manoj three times as old as Ram?
Q22.
Find three consecutive even numbers whose sum is 96.
Q23.
Find three consecutive numbers such that the sum of the second and the third
numbers exceeds the first by 14.
Q24.
Dad is seven times as old as his son. 10 years later, he will be three times as old
as his son. Find their ages.
Q25.
Divide 64 into two parts such that three times the greater part will be equal to
five times the smaller one.
87
Suggested Video Links and Resources
Name
Title/Link
Video Clip 1
Writing Basic Algebraic Expression
http://www.youtube.com/watch?v=3BXmzyhYcf8
Video Clip 2
Evaluating Algebraic Expression
http://www.youtube.com/watch?v=vQT0e_p_Z8s&feature=relmfu
Video Clip 3
Simplifying of Algebraic Expression
http://www.youtube.com/watch?v=_A8lLbZCrlw&feature=related
Video Clip 4
Simple Equations
http://www.youtube.com/watch?v=9Ek61w1LxSc
Video Clip 5
Solving Simple Equations
http://www.youtube.com/watch?v=3PAZs0R0yfk
Video Clip 6
Writing Algebraic Expression 1 (choose 9.9)
http://website-tools.net/google-keyword/site/kentbutler.pbworks.com
Video Clip 7
Writing Algebraic Expression 2
http://schools.paulding.k12.ga.us/ischooldistrict/media/files/1954/Writing
%20Algebraic%20Expressions.ppt
Weblink1
http://www.math-play.com/Algebraic-Expressions-Millionaire/algebraicexpressions-millionaire.html
Weblink 2
http://www.quia.com/mc/319817.html
Weblink 3
http://algebra4children.com/Games/games-2/Algebra-define-variable6/hoopshoot-algebra-define-variable-6.html
Weblink 4
http://algebra4children.com/Games/games-2/Algebra-define-variable1/walk%20plank%20define%20variable%201.html
Weblink 5
http://www.aplusmath.com/Games/PlanetBlast/index.html
Weblink 6
http://infinity.cos.edu/webmath2/default.asp
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