NCERT Solutions: Linear Equations in One Variable (Exercise 2.2)

Exercise: 2.2

Q1. 

Ans: Transposing to RHS and x/3  to LHS, we have

(Multiplying both sides by 6) 

Check:   

Q2.    
Ans: ∵  LCM of 2, 4 and 6 = 12 
∴  Multiplying both sides by 12, we have

or 
6n - 9n + 10n = 252
or
7n = 252
or  
n = 252/7 = 36 
∴  n = 36
Check: 

∴ LHS = RHS

Q3. 
Ans: ∵ LCM of 3, 6 and 2 is 6. 
∴ Multiplying both sides by 6, we have

or 6x + 42 - 16x = 17 - 15x
or (6 - 16)x + 42 = 17 - 15x
or -10x + 42 = 17 - 15x
Transposing 42 to RHS and -15x to LHS, we have
-10x + 15x = 17 - 42 or 5x = -25
or 
5x = -25
or 
x = -25/5 = -5    (Dividing both sides by 5)
∴ x  = -5
Check:
    


∴ LHS = RHS

Q4. 
Ans: ∵ LCM of 3 and 5 is 15.
∴ Multiplying both sides by 15, we have

or 
5(x - 5) = 3(x - 3)
or
5x - 25 = 3x - 9
Transposing (-25) to RHS and 3x to LHS, we have
5x - 3x = -9 + 25
or
2x = 16
or 
x = 16/2   (Dividing both sides by 2)
∴ x = 8
Check:
= 3/3 = 1


∴ LHS = RHS

Q5. 
Ans: ∵  LCM of 4 and 3 is 12.
∴  Multiplying both sides by 12, we have

or 
3(3t - 2) - 4(2t + 3) = (4 x2) - 12t
or  
9t - 6 - 8t - 12 = 8 - 12t
or  
(9 - 8)t - (6 + 12) = 8 - 12t
or  
t - 18 = 8 - 12t
Transposing -18 to RHS and -12t to LHS, we have
t + 12t = 8 + 18
or
13t = 26
or
t = 26/13 
∴ t = 2
Check:


∴ LHS = RHS

Q6.  
Ans: Since, LCM of 2 and 3 is 6.
∴   Multiplying both sides by 6, we have

or
6m - 3(m - 1) = 6 - 2(m - 2)
or
6m - 3m + 3 = 6 - 2m + 4
or
(6 - 3)m + 3 = (6 + 4) - 2m
or
3m + 3 = 10 - 2m
Transposing 3 to RHS and -2m to LHS, we have
3m + 2m = 10 - 3
or  5m = 7
or 
m = 7/5    (Dividing both sides by 5)
Check:



∴ LHS = RHS

Simplify and solve the following linear equations.
Q7. 3(t - 3) = 5(2t + 1)
Ans: 3(t - 3) = 5(2t + 1)
⇒ 3t - 9 = 10t + 5
⇒ 3t - 10t = 5 + 9
⇒ -7t = 14
⇒ t = 14/-7
⇒ t = -2

Q8. 15(y - 4) - 2(y - 9) + 5(y + 6) = 0
Ans: 15(y - 4) -2(y - 9) + 5(y + 6) = 0
⇒ 15y - 60 -2y + 18 + 5y + 30 = 0
⇒ 15y - 2y + 5y = 60 - 18 - 30
⇒ 18y = 12
⇒ y = 12/18
⇒ y = 2/3

Q9. 3 (5z - 7) - 2(9z - 11) = 4(8z - 13) - 17
Ans: 3(5z - 7) - 2(9z - 11) = 4(8z - 13) - 17
⇒ 15z - 21 - 18z + 22 = 32z - 52 - 17
⇒ 15z - 18z - 32z = -52 - 17 + 21 - 22
⇒ -35z = -70
⇒ z = -70/-35
⇒ z = 2

Q10. 0.25(4f - 3) = 0.05(10f - 9)
Ans: 0.25(4f - 3) = 0.05(10f - 9)
⇒ f - 0.75 = 0.5f - 0.45
⇒ f - 0.5f = -0.45 + 0.75
⇒ 0.5f = 0.30
⇒ f = 0.30/0.5
⇒ f = 3/5
⇒ f = 0.6

Old NCERT Questions

Solve the following equations
1.
2.
3.
4.
5.
Ans:  
1.
Multiplying both sides by 3x, we have

or
8x - 3 = 6x 
Transposing (-3) to RHS and 6x to LHS, we have 
8x - 6x = 3
or
2x = 3 
Dividing both sides by 2, we have
x = 3/2
2.
Multiplying both sides by 7 - 6x, we have

or
9x = 105 - 90x 
Transposing (-90x) to LHS, we have 
9x + 90x = 105
or
99x = 105
or
x = 105/99 (Dividing both sides by 99)
or
x = 35/33
3. 
By cross multiplication, we have
9z = 4(z + 15) ⇒ 9z = 4z + 60 
Transposing 4z to LHS, we have
9z - 4z = 60  
5z = 60 ⇒  z = 60/5 = 12
∴   z = 12
4. 
By cross multiplication, we have
5(3y + 4) = -2(2 - 6y)
or 
15y + 20 = -4 + 12y
Transposing 20 to RHS and 12y to LHS, we have
15y - 12y = -4 - 20
or
3y = -24
or
y = - 24/3= -8    (Dividing both sides by 3)
or   y = -8
5. 
By cross multiplication, we have
3 * [7y + 4] = -4 x [y + 2]
or 
21y + 12 = -4y - 8
Transposing 12 to RHS and (-4y) to LHS, we have
21y + 4y = -8 - 12
or
25y = -20
or
y = -20/25 (Dividing both sides by 25)
or  
y = -20/25 = -4
∴   y = -4/5

Question 6: The ages of Hari and Harry are in the ratio 5: 7. Four years from now the ratio of their ages will be 3: 4. Find their present ages.
Ans: Let the present age of Hari = 5x years
and the present age of Harry = 7x years
After 4 years, Age of Hari = (5x + 4) years
Age of Harry = (7x + 4) years 
According to the condition, 
(5x + 4) : (7x + 4) = 3 : 4
or
    
By cross multiplication, we have: 
4(5x + 4) = 3(7x + 4)
or
20x + 16 = 21x + 12 
Transposing 16 to RHS and 21x to LHS, we have 
20x - 21x = 12 - 16 
-x = -4 ⇒ x = 4
∴ Present age of Hari = 5 * 4 = 20 years
Present age of Harry = 7 * 4 = 28 years

Question 7: The denominator of a rational number is greater than its numerator by 8. If the numerator is increased by 17 and the denominator is decreased by 1, the number obtained is 3/2. Find the rational number.
Ans: Let the numerator = x 
∴ Denominator = x + 8
New numerator = (x) + 17
New denominator = (x + 8) - 1 = x + 7
∴ The new number =
According to the condition, we have 

By cross multiplication, we have
2(x + 17) = 3(x + 7) 2x + 3x = 3x + 21 
Transposing 34 to RHS and 3x to LHS, we have 
2x - 34 = 21 - 34 ⇒ -x = -13
∴ x= 13 ⇒ Numerator = 13 
x + 8 = 13 + 8 = 21 ⇒Denominator = 21
∴ The rational number = 13/21