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Finding the value of a variable in a linear equation that Satisfies the equation is called a root of the equation.

(i) solution

(ii) 1,

(iii) variables

False.

Consider the equation, ((5x – 7)/2) = y

Then, from the question it is given that the value of x = 9

So, (((5 × 9) – 7)/2) = y

((45 – 7)/2) = y

38/2 = y

y = 19

(c) – 6

Consider the given equation –6 + x = –12.

Substitute the value of x,

-6 + (-6) = -12

-12 = -12

Left hand side is equal to Right hand side.

(a) If smaller number is x, the other number is .(use sum)

From the question it is given that smaller number is x,

Then, the other number is 60 – x

(b) The difference of numbers in term of x is .

From the question, smaller number be x, and other number be (60 – x)

Then,

Difference between the numbers = (60 – x) – x

= 60 – x – x

= 60 – 2x

(c) The equation formed is .

As per the condition given the question, differenc of the two numbers is 30.

So, 60 – 2x = 30

Transpose, -2x to RHS then it becomes 2x and 30 to LHS it becomes -30,

60 – 30 = 2x

30 = 2x

(d) The solution of the equation is .

30 = 2x

x = 30/2

x = 15

(e) The numbers are and .

x = 15

60 – x = 60 – 15

= 45

The numbers are 15 and 45.

Correct answer is (a) 0.002

Correct answer is 18

Let the number be x.

So, half of the number = x/2

On adding 45 to it, we get x/2 + 45

x/2 + 45 = 3x

3x - x/2 = 45

(6x - x)/2 = 45

5x = 45 x 2 = 90

x = 90/5 = 18

10 less than a number is 65 = x -10

Then, the number is x – 10 = 65

Transpose, – 10 to RHS then it becomes 10

x = 65 + 10

x = 75

(a) 0.002

Consider the given equation 43m = 0.086 to find out the value of m.

m = 0.086/43

m = 0.002

Let us assume the number be x.

If a number is increased by 20 = x + 20

If a number is increased by 20, it becomes 45, x + 20 = 45

Transpose, 20 to RHS then it becomes -20.

x = 45 – 20

x = 25

Correct answer is (a). 

x = 5 on multiplying both sides by 2 gives 2x = 10 which on adding 3 both sides gives 2x + 3 =13

Let us assume the number be x.

If a number is increased by 12 = x + 12

If a number is increased by 12, it becomes 84, x + 12 = 84

Transpose, 12 to RHS then it becomes -12.

x = 84 – 12

x = 72

Let Radha’s monthly salary = ₹ x

So, money got by her in over time = ₹ (17480 – x)

According to question,

x = 17480 – x + 10000

⇒ x + x = 17480 + 10000

⇒ 2x = 27480

⇒ x = 27480/2 = 13740

Thus, 13740 is her monthly salary.

(c) x exceeds 3 by 7, can be represented as x – 3 = 7.

19

2x + 19 = 11

2x = 11 - 19

2x = -8

x = -8/2

x = -4

22

y - 13 = 9

y = 9 +13

y = 22

Let the number be x.

So, one-fifth of number = x/5

Therefore, x/5 = x - 5 is the required equation.

In equation x/2 – 2 = 4, the value of x will be 12.

Let the number be x. 

So, twice of number – 2x

On subtracting 13 from it, we get 2x – 13 

Therefore, 2x – 13 = 3 is the required equation.

(i) 4x + x – 13 = 7 put x = 4

L.H.S. = 4x + x – 13 = 5x – 13
= 5 x 4 – 13
= 20 – 13 = 7 = RHS

Hence 4x + x -13 = 7 put x = 4 is correct.

(ii) 3x – 8 = 25 put x = 12

LHS = 3 – 8
= 3 x 12 – 8
= 36 – 8 = 28 ≠ RHS

Hence 3x – 8 = 25 put x = 12 is in correct.

(iii) 7x – 5 = 3x +7 put x = 3

7x – 5 = 3x +7
7 x 3 – 5 = 3 x 3 + 7
21 – 5 = 9 + 7
16 = 16
LHS = RHS

Hence 7x – 5 = 3x +7 put x = 3 is correct.

(iv) 5x – 7 = 4x +1 put x = 5

LHS or RHS put x= 5
5 x 5 – 7 = 4 x 5 + 1
25 – 7 = 20 + 1
18 ≠ 21
LHS ≠ RHS

Hence 5x – 7 = 4x + 1 put x = 5 is correct.

(d) 3 + x = 8

Consider the given equation 3 + x = 8.

Transpose 3 to right hand side then it becomes -3

x = 8 – 3

x = 5

(c) 10 + 3x = 1

Consider the given equation 10 + 3x = 1.

Transpose 10 to right hand side then it becomes -10

3x = 1 – 10

x = -9/3

x = -3

(i) Let Parmits marble be x

According to the question

5x + 7 = 37

5x = 37 – 7 = 30

x = \(\frac{30}{5} = 6\)

∴ Parmits marble be = x = 6

(ii) Let the age of Laxmi be ‘x’ years

According to the question

3x + 4 = 49

3x = 49 – 4 = 45

3x = 45

x = 45/3 = 15

∴ Lakshmi’s age be x = 15 years.

(iii) Let the number of fruit trees be ‘x’

According to the question

3x + 2 = 77

3x = 77 – 2 = 75

x = \(\frac{75}{3} = 25\)

∴ No. of fruit trees = x = 25

(i) True

(ii) False

(iii) False

(iv) True

(i)  Let the number of marbles Parmit had be ‘m’
∴ Number of marbles Irfan has = Five times of parmits marble 5m.
To this 7 marbles more means = 5m + 7
∴ 5m + 7 = 37.

(ii) Let the age of Lakshmi be ‘y’ years
Three times of age = 3y
More than 4 years = 3y + 4
∴ Lakshmi’s father age be = 3y + 4
∴ 3y + 4 = 49

(iii)  Let the lowest score be ‘l’
Twice the lowest score be = 2l
Plus seven = 2l + 1
∴ The highest score = 2l + 7
∴ 2l + 7 = 87

(iv) Let the base angle be b°
Vertex angle = Twice the base angle
∴ Vertex angle = 2b
Sum of the angles of an isosceles triangle is =180°
∴ 2b + b + b = 180° = 4b = 180°

(a) 1 + 5 ≠ 19
6 ≠ 19
∴ LHS ≠ RHS to
∴ n = 1 is not a solution to the given equation n + 5 = 19.

(b) 7 × -2 + 5 = 19
-14 + 5 = 19
-9 = 19
LHS ≠ RHS
∴ n = -2 is not a solutions the given equation 7n + 5 = 19 (n = 2)
7 × 2 + 5 = 19
14 + 5 = 19
19 = 19
LHS = RHS
∴ n = 2 is a solution to the given equation 7n + 5 = 19

(c) 7 × 2+5 = 19
14 + 5 = 19
19 = 19
LHS = RHS
∴ n = 2 is a solution to the given equation 7n + 5 = 19.

(d) 4(1) – 3 = 13
4 – 3 = 13
1 ≠ 13
LHS ≠ RHS
∴ p = 1 is not a solution to the given equation 4p – 3 = 13.

(e)  4(-4) – 3 = 13
– 16 – 3 = 13
-19 ≠ 13
LHS ≠ RHS
∴ p = -4 is not a solution to the given equation 4p – 3 = 13.

(f)  4(0) – 3 = 13
0 – 3 = 13
– 3 ≠ 13
LHS ≠ RHS
∴ p = 0 is not a solution to the given equation 4p – 3 = 13.

(c) 3x – 1 = – 1

Consider the given equation 3x – 1 = – 1.

Transpose -1 to right hand side then it becomes 1

3x = 1 + 1

x = 0/3

x = 0

(i) The sum of P and 4 is equal to 15 .

(ii) Seven subtracted from m is equal to 3.

(iii) Two times a number m is equal to 7.

(iv) One – fifth of ‘m’ number is equal to 3m

(v)Three – fifth of ‘m’ number is equal to 6

(vi) Three times a number ‘P’ is added to 4 is equal to 25.

(vii) Two is subtracted from 4 times ‘p’ is equal to 18.

(viii) Two is added to half number of p is equal to 8.v

(d) 4x + 7 = x + 2

Consider the equation 4x + 7 = x + 2

Transpose 7 to right hand side then it becomes –7 and x to Left hand side then it becomes – x.

4x – x = 2 – 7

3x = -5

x = -5/3

Correct answer is (a) Adding the same number to both sides of the equation.

(i) x + 4 = 9

(ii) y – 2 = 8

(iii) 10a = 70

(iv) \(\frac{3}{4t} = 15\)

(v) t = 15

(vi) 7m + 1 = 11

(vii) \(\frac{1}{4}x - 4 = 4\)

(viii)  6y – 6 = 60

(ix) \(\frac{1}{3}\)z + 3 = 30

(c) –b/a

Consider the given equation ax + b = 0

Transpose ‘b’ to right hand side then it becomes –b.

ax = -b

x = -b/a

(d) As dividing both sides of the equation by a non-zero number is allowed.

(a) positive number

The solution of the given equation ax = b

x = b/a

(i) If p = 0 then 5(0) + 2 = 17, 0 + 2 ≠ 17
If p = 1 then 5(1) + 2 = 17, 5 + 2 ≠ 17
If p = 2 then 5(2) + 2 = 17, 10 + 2 ≠ 17
If p = 3 then 5(3) + 2 = 17, 15 + 2 ≠ 17
∴ p = 3 is the solution to the given equation 5p + 2 = 17.

(ii) If m = 0 then 3(0) – 14
0 – 14 = – 14 ≠ 4
3m – 14 = 4
If m = 1 then 3(1) – 14
3 – 14 = -11
If m = 2 then 3(2) – 14
6 – 14 = – 8
If m = 3 then 3(3) – 14 = -5
If m = 4 then 3(4) – 14 = -2
If m = 5 then 3(5) – 14 = 1
If m = 6 then 3(6) – 14 = 4
LHS = RHS
∴ m = 6 is the solution to the given equation 3m – 14 = 4.

True.

Consider the eqaution, 3x + 2 = 17

Transpose 2 to RHS then it becomes – 2.

3x = 17 – 2

3x = 15

x = 15/3

x = 5

Let the number be x.

So, two times of the number = 2x

When, increased by 5, it gives the expression 2x + 5

∴ 2x + 5 = 9

⇒ 2x = 9 – 5 = 4

⇒ x = 4/2 = 2

Thus, x = 2 is the required number.

Let the required number be x. 

So, 2 times this number = 2x 

When increased by 5, it gives the expression 2x + 5 

Thus, required equation is 2x + 5 = 9.

False.

Consider the eqaution, 4x – 7 = 11

Transpose -7 to RHS then it becomes 7.

4x = 11 + 7

4x = 18

x = 18/4

x = 9/2

False.

Consider the eqaution, 4x – 1 = 8

Transpose -1 to RHS then it becomes 1.

4x = 8 + 1

4x = 9

x = 9/4

Let the number be x.

So, twice of the number = 2x

On adding 4 to it, we get 2x + 4

∴ 2x + 4 = 18 is the required equation.

True

x + 2 = 5   .......(1)

x = 5 - 2

x = 3

3x - 1 = 8 .........(2)

3x = 8 + 1

3x = 9

x = 9/3

x = 3

Eqs. (1) and (2) are same.

False 

LHS = 4 × 12 – 5 = 43 and 

RHS = 3 × 12 + 10 = 46 

They are not equal.

False.

A number x divided by 7 gives 2 can be written as (x +1)/7 = 2

Let present age of Manoj be x years.

So, five times of his age = 5x

According to question,

5 = x + 20

⇒ 5x – x = 20

⇒ 4x = 20

⇒ x = 20/4 = 5

Thus, at present Manoj is 5 years old.

Let the number be x. 

According to the given condition,

1/ 3 x – 1 = 1 

or 1/3 x = 1 + 1 

or 1/3 x = 2 

or x = 6.

Let the number be x.

On subtracting 1 from it, we get x – 1

Multiplying it by 1/2 we get 1/2(x-1)

1/2(x-1) = 7 is the required equation.

Correct answer is 6