06.11 Maths
Ch. 11. Algebra
लेखक
ॐ जितेन्द्र सिंह तोमर
M.A., B.Ed., DNYS, MASSCOM.
10/4/10/2/2022
Arithmetic
The branch of mathematics in which we studied numbers is arithmetic.
Geometry
The branch of mathematics in which we studied about various shapes is geometry.
Algebra
The branch of mathematics in which we represents and studies numbers with digits, letters and symbols is called algebra.
गणित की वह शाखा जिसमें अंक, वर्णों एवं प्रीतीकों के माध्यम से संख्याओं को दर्शाया जाता है; बीजगणित कहलाता है।
The Idea of a Variable
To understand the idea of a variable we are going to make a patterns with matchsticks. We decide to make simple patterns of the letters of the English alphabet ' L '.
* We takes two matchsticks and forms the letter L as shown in Fig (a).
* We again picks two sticks, forms another letter L and puts it next to
the one made earlier like (b).
We again picks two sticks, forms another letter L and puts it next to
the one made earlier like (c).
When I looks at the pattern. I want to asks the question, “How many matchsticks will be required to make seven Ls”?
Now you will go on forming the patterns with 1L, 2Ls, 3Ls, and so on.
Number of Number of Total
Ls formed matchsticks
required
1 L 1 × 2 2
2 L 2 × 2 4
3 L 3 × 2 6
4 L 4 × 2 8
5 L 5 × 2 10
n L n × 2 2n
The rule is :
Number of matchsticks required = 2n
= 2 × No. of Ls.
Q. Suppose we want to make ten Ls, how many matchsticks we needs?
Sol.
No. of Ls (n) = 10
Number of matchsticks required = 2n
= 2 × No. of Ls.
= 2 × 10
= 20 matchsticks.
Here ' n ' is an example of a variable. Its value is not fixed; it can take any value 1, 2, 3, 4, ... .
We can write the rule for the number of matchsticks required using the variable n.
Variable
The word ‘variable’ means something that can vary, i.e. change. The value of a variable is not fixed. It can take different values.
Thus the Variables means changeable or not fix.
Remember, a variable is a number which does not have a fixed value. We may use any letter as m, l, p, x, y, z etc. to show a variable.
Make Other Patterns
For C
Number of Number of Total
Ls formed matchsticks
required
1 L 1 × 3 3
2 L 2 × 3 6
3 L 3 × 3 9
4 L 4 × 3 12
5 L 5 × 3 15
n L n × 3 3n
The rule is :
Number of matchsticks required = 3n
= 3 × No. of Ls.
The expression (x + 10) cannot be simplified further.
Do not confuse x + 10 with 10x, they are different.
x + 10 ≠ 10x
In 10x, x is multiplied by 10 and In (x + 10), 10 is added to x.
We may check this for some values of x.
For example,
If x = 2, then
10x = 10 × 2 = 20 and for
x + 10 = 2 + 10 = 12.
If x = 10, then
10x = 10 × 10 = 100 and for
x + 10 = 10 + 10 = 20.
Find the rule which gives the number of matchsticks required to make the
following matchstick patterns. Use a variable to write the rule.
(a) A pattern of letter A
(b) A pattern of letter C
(c) A pattern of letter E
(d) A pattern of letter F
(e) A pattern of letter H
(f) A pattern of letter L
(g) A pattern of letter N
(h) A pattern of letter S
(i) A pattern of letter T
(j) A pattern of letter U
(k) A pattern of letter V
(l) A pattern of letter Z
Make Other Patterns
For O
Number of Number of Total
Ls formed matchsticks
required
1 O 1 × 3 + 1 4
2 O 2 × 3 + 1 7
3 O 3 × 3 + 1 10
4 O 4 × 3 + 1 13
5 O 5 × 3 + 1 16
n O n × 3 + 1 3n + 1
The rule is :
Number of matchsticks required = 3n + 1
= 3 × No. of Os. + 1
Make Other Patterns
For ∆
Number of Number of Total
Ls formed matchsticks
required
1 ∆ 1 × 2 + 1 3
2 ∆ 2 × 2 + 1 5
3 ∆ 3 × 2 + 1 7
4 ∆ 4 × 2 + 1 9
5 ∆ 5 × 2 + 1 11
n ∆ n × 2 + 1 2n + 1
The rule is :
Number of matchsticks required = 2n + 1
= 2 × No. of ∆s. + 1
Use of Variables in Common Rules
Perimeter
We know that perimeter of any polygon (a closed figure made up of 3 or more line segments) is the sum of the lengths of its sides.
p = sum of the lengths of its sides
Perimeter Of A Square
A square has 4 sides and they are equal in length. Therefore,
The perimeter of a square = Sum of the lengths of all the sides of the square
perimeter (p) = 4 × length of side (l)
p = 4 × l
p = 4l.
This rule expressed a relation between the perimeter and the length of the square.
Perimeter Of A Rectangle
AB = CD = l
BC = AD = b
If we denote by l, the length of the sides AB or CD and, by b, the length of the sides AD or BC.
Therefore,
Perimeter of a rectangle (p) = length of AB (l) + length of BC (b) + length of CD (l) + length of AD (b)
p = l + b + l + b
p = 2 × l + 2 × b
p = 2l + 2b
The rule, for the perimeter of a rectangle is p = 2l + 2b
where, l and b are respectively the length and breadth of the rectangle.
and p the perimeter of the rectangle
Commutativity Of Addition Of Two Numbers
We know that
4 + 3 = 7 and 3 + 4 = 7
i.e. 4 + 3 = 3 + 4
a + b = b + a
Commutativity Of Multiplication Of Two Numbers
We know that
4 × 3 = 12 and 3 × 4 = 12
i.e. 4 × 3 = 3 × 4
a × b = b × a
Distributivity Of Multiplication Over Addition and Subtraction Of Numbers
a × (b + c) = a × b + a × c
a × (b – c) = a × b – a × c
We know that variables can take different values; they have no fixed
value. But actually variables are numbers. That is why as in the case of numbers, operations of addition, subtraction, multiplication and division can be done on them.
One important point must be noted regarding the expressions containing variables. A number expression like (4 × 3) + 5 can be immediately evaluated as (4 × 3) + 5 = 12 + 5 = 17
But an expression like (4x + 5), which contains the variable x, cannot be evaluated. Only if x is given some value, an expression like (4x + 5) can
be evaluated. For example, when x = 3, 4x + 5 = (4 × 3) + 5 = 17 as found above.
Expression How formed?
(a) y + 5 5 added to y
(b) t – 7 7 subtracted from t
(c) 10 a a multiplied by 10
(d)x/3 x divided by 3
(e) – 5 q q multiplied by –5
(f) 3 x + 2 first x multiplied by 3,
then 2 added to the
product
(g) 2 y – 5 first y multiplied by 2,
then 5 subtracted
from the product
Give expressions for the following :
(a) 12 subtracted from z z – 12
(b) 25 added to r r + 25
(c) p multiplied by 16 16 p
(d) y divided by 8 y/8
(e) m multiplied by –9 – 9 m
(f) y multiplied by 10 10 y + 7
and then 7 added to
the product
(g) n multiplied by 2 and 2 n – 1
1 subtracted from the product
Give expressions for the following cases.
(a) 7 added to p
(b) 7 subtracted from p
(c) p multiplied by 7
(d) p divided by 7
(e) 7 subtracted from – m
(f) – p multiplied by 5
(g) – p divided by 5
(h) p multiplied by – 5
Give expressions in the following cases.
(a) 11 added to 2m
(b) 11 subtracted from 2m
(c) 5 times y to which 3 is added
(d) 5 times y from which 3 is subtracted
(e) y is multiplied by – 8
(f) y is multiplied by – 8 and then 5 is added to the result
(g) y is multiplied by 5 and the result is subtracted from 16
(h) y is multiplied by – 5 and the result is added to 16.
Ans.
4. (a) p + 7
(b) p – 7
(c) 7 p
(d) p/7
(e) – m – 7
(f) – 5p
(g)–p/5
(h) – 5 p
5. (a) 2m + 11
(b) 2m – 11
(c) 5y + 3
(d) 5y – 3
(e) – 8y
(f) – 8 y + 5
(g) 16 – 5y
(h) – 5y + 16
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