NCERT Solutions for class 8 Maths Chapter 2: Linear Equations in one variable

Exercise 2.1

Solve the following equations and check your results.

1. 3x = 2x + 18

Answer: x = 18

Explanation: 3x = 2x + 18

Subtracting 2x from both the sides,

3x - 2x = 2x + 18 - 2x

3x - 2x = 2x- 2x + 18

x = 0 + 18

x = 18

2. 5t - 3 = 3t - 5

Answer: t = - 1

Explanation: 5t - 3 = 3t - 5

Transposing 3t to the LHS (Left Hand Side),

5t - 3 - 3t = - 5

5t - 3t - 3 = - 5

2t - 3 = - 5

Transposing (- 3) to the RHS (Right Hand Side),

2t = - 5 + 3

2t = - 2

Dividing by 2,

2t/2 = - 2/2

t = - 1

Note: Transposing of a number always results in the change of the sign of that number.

3. 5x + 9 = 5 + 3x

Answer: x = -2

Explanation: 5x + 9 = 5 + 3x

Transposing 3x to the LHS (Left Hand Side),

5x + 9 - 3x = 5

5x - 3x + 9 = 5

2x + 9 = 5

Transposing (9) to the RHS (Right Hand Side),

2x = 5 - 9

2x = -4

Dividing by 2,

2x/2 = -4/2

x = -2

4. 4z + 3 = 6 + 2z

Answer: z = 3/2

Explanation: 4z + 3 = 6 + 2z

Transposing 2z to the LHS,

4z + 3 - 2z = 6

4z - 2z + 3 = 6

2z + 3 = 6

Subtracting 3 from both the sides,

2z + 3 - 3 = 6 - 3

2z + 0 = 3

2z = 3

Dividing by 2,

2z/2 = 3/2

z = 3/2

5. 2x - 1 = 14 - x

Answer: x = 5

Explanation: 2x - 1 = 14 - x

Transposing - x to LHS,

2x - 1 + x = 14

2x + x - 1 = 14

3x - 1 = 14

Adding 1 to both the sides,

3x - 1 + 1 = 14 + 1

3x + 0 = 15

3x = 15

Dividing by 3,

3x/3 = 15/3

x = 5

6. 8x + 4 = 3 (x - 1) + 7

Answer: x = 0

Explanation: 8x + 4 = 3 (x - 1) + 7

Let's open the brackets.

8x + 4 = 3x - 3 + 7

8x + 4 = 3x + 4

Transposing 3x to the LHS,

8x - 3x + 4 = 4

5x + 4 = 4

Subtracting 4 from both the sides,

5x + 4 - 4 = 4 - 4

5x + 0 = 0

5x = 0

x = 0

7. x = 4/5 (x + 10)

Answer: x = 40

Explanation: x = 4/5 (x + 10)

Let's open the brackets first on right side.

x = 4x/5 + 4/5 × 10

x = 4x/5 + 8

Multiplying the equation by 5,

5x = 5 × (4x/5 + 8)

5x = 4x + 40

Transposing 4x to LHS,

5x - 4x = 40

x = 40

8. 2x/3 + 1 = 7x/15 + 3

Answer: x = 10

Explanation: 2x/3 + 1 = 7x/15 + 3

Multiplying the equation by 15,

15 × (2x/3 + 1) = 15 × (7x/15 + 3)

10x + 15 = 7x + 45

Subtracting 15 from both the sides,

10x + 15 - 15 = 7x + 45 - 15

10x + 0 = 7x + 30

10x = 7x + 30

Transposing 7x to LHS,

10x - 7x = 30

3x = 30

Dividing by 3,

3x/3 = 30/3

x = 10

9. 2y + 5/3 = 26/3 − y

Answer: y = 7/3

Explanation: 2y + 5/3 = 26/3 - y

Multiplying the equation by 3,

3 × (2y + 5/3) = 3 × (26/3 - y)

6y + 5 = 26 - 3y

Transposing (- 3y) to LHS,

6y + 3y + 5 = 26

9y + 5 = 26

Subtracting 5 from both the sides,

9y + 5 - 5 = 26 - 5

9y = 21

Dividing by 9,

9y/9 = 21/9

y = 21/9

y = 7/3

10. 3m = 5 m - 8/5

Answer: m = 4/5

Explanation: 3m = 5 m - 8/5

Transposing 5m to LHS,

3m - 5m = - 8/5

- 2m = - 8/5

Negative sign on the both sides will be cancelled.

2m = 8/5

Multiplying the equation by 5,

2m × 5 = 8/5 × 5

10m = 8

Dividing by 10,

10m/10 = 8/10

m = 8/10

m = 4/5

We can check LHS and RHS by substituting the value of the variable in the equation.

Exercise 2.2

Solve the following linear equations.

1. x/2 - 1/5 = x/3 + 1/4

Answer: x = 27/10

Explanation: Multiplying the equation by 60,

60 × (x/2 - 1/5) = 60 × (x/3 + 1/4)

30x - 12 = 20x + 15

Transposing 20x to LHS,

30x - 20x - 12 = 15

10x - 12 = 15

Adding 12 to both the sides,

10x - 12 + 12 = 15 + 12

10x + 0 = 27

10x = 27

Dividing by 10,

10x/10 = 27/10

x = 27/10

2. n/2 - 3n/4 + 5n/6 = 21

Answer: n = 36

Explanation: Multiplying the equation by 12,

12 × (n/2 - 3n/4 + 5n/6) = 12 × 21

6n - 9n + 10n = 252

7n = 252

Dividing by 7,

7n/7 = 252/7

n = 36

3. x + 7 - 8x/3 = 17/6 - 5x/2

Answer: x = - 5

Explanation: Multiplying the equation by 6,

6 × (x + 7 - 8x/3) = 6 × (17/6 - 5x/2)

6x + 42 - 16x = 17 - 15x

42 - 10x = 17 - 15x

Transposing (- 15x) to LHS,

42 - 10x + 15x = 17

42 + 5x = 17

Subtracting 42 from both the sides,

42 + 5x - 42 = 17 - 42

5x + 0 = - 25

5x = - 25

Dividing by 5,

5x/5 = - 25/5

x = - 5

4. (x - 5)/3 = (x - 3)/5

Answer: x = 8

Explanation: Multiplying the equation by 15,

15 × ((x - 5)/3) = 15 × ((x - 3)/5)

5(x - 5) = 3 (x - 3)

Opening the brackets,

5x - 25 = 3x - 9

Transposing 3x to LHS,

5x - 3x - 25 = - 9

2x - 25 = - 9

Adding 25 to both the sides,

2x - 25 + 25 = - 9 + 25

2x = 16

Dividing by 2,

2x/2 = 16/2

x = 8

5. (3t - 2)/4 - (2t + 3)/3 = 2/3 - t

Answer: t = 2

Explanation: Multiplying the equation by 12,

12 × ((3t - 2)/4 - (2t + 3)/3) = 12 × (2/3 - t)

3 (3t - 2) - 4 (2t + 3) = 8 - 12t

9t - 6 - 8t - 12 = 8 - 12t

t - 18 = 8 - 12t

Transposing (- 12t) to LHS,

t - 18 + 12t = 8

13t - 18 = 8

Adding 18 to both the sides,

13t - 18 + 18 = 8 + 18

13t + 0 = 26

13t = 26

Dividing by 13,

13t/13 = 26/13

t = 2

6. m - (m - 1)/2 = 1 - (m - 2)/3

Answer: m = 7/5

Explanation: Multiplying the equation by 6,

6 × (m - (m - 1)/2) = 6 × (1 - (m - 2)/3)

6m - 3(m - 1) = 6 - 2(m - 2)

6m - 3m + 3 = 6 - 2m + 4

3m + 3 = 10 - 2m

Transposing (- 2m) to PHS,

3m + 3 + 2m = 10

5m + 3 = 10

Subtracting 3 from both the sides,

5m + 3 - 3 = 10 - 3

5m + 0 = 7

5m = 7

Dividing by 5,

5m/5 = 7/5

m = 7/5

Simplify and solve the following linear equations.

7. 3(t - 3) = 5(2t + 1)

Answer: t = - 2

Explanation: 3(t - 3) = 5(2t + 1)

Let's open the brackets.

3t - 9 = 10t + 5

Transposing 10t to LHS,

3t - 10t - 9 = 5

- 7t - 9 = 5

Adding 9 to both the sides,

- 7t - 9 + 9 = 5 + 9

- 7t + 0 = 14

- 7t = 14

Dividing by - 7,

- 7t/-7 = 14/-7

t = - 14/7

t = - 2

8. 15(y - 4) -2(y - 9) + 5(y + 6) = 0

Answer: y = 2/3

Explanation: 15(y - 4) -2(y - 9) + 5(y + 6) = 0

The equation after opening the brackets becomes,

15y - 60 - 2y + 18 + 5y + 30 = 0

15y - 2y + 5y - 60 + 18 + 30 = 0

18y - 12 = 0

Adding 12 to both the sides,

18y - 12 + 12 = 0 + 12

18y + 0 = 12

18y = 12

Dividing by 18,

y = 12/18

y = 2/3

9. 3(5z - 7) - 2(9z - 11) = 4(8z - 13) - 17

Answer: z = 2

Explanation: 3(5z - 7) - 2(9z - 11) = 4(8z - 13) - 17

The equation after opening the brackets becomes,

15z - 21 - 18z + 22 = 32z - 52 - 17

- 3z + 1 = 32z - 69

Transposing - 3z to RHS,

1 = 32z + 3z - 69

1 = 35z - 69

Adding 69 to both the sides,

1 + 69 = 35z - 69 + 69

70 = 35z + 0

70 = 35z

Dividing by 35,

z = 70/35

z = 2

10. 0.25(4f - 3) = 0.05(10f - 9)

Answer: f = 0.6

Explanation: 0.25(4f - 3) = 0.05(10f - 9)

The equation after opening the brackets becomes,

f - 0.75 = 0.5f - 0.45

Transposing 0.5z to LHS,

f - 0.5 f - 0.75 = - 0.45

0.5f - 0.75 = - 0.45

Adding 0.75 to both the sides,

0.5f - 0.75 + 0.75 = - 0.45 + 0.75

0.5f + 0 = 0.30

0.5f = 0.30

Dividing by 0.5,

0.5f/0.5 = 0.30/0.5

f = 0.6






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