Solve each of the following equations and verify the answer in each case:

Question 1:

x + 5 = 12

Solution:

x + 5 = 12

Subtracting 5 from both the sides:

\(\Rightarrow\)

\(\Rightarrow\)

Verification:

Substituting x = 7 in the L.H.S.

\(\Rightarrow\)

L.H.S. = R.H.S.

Hence, verified.

Question 2:

x + 3 = -2

Solution:

x + 3 = -2

Subtracting 3 from both the sides:

\(\Rightarrow\)

\(\Rightarrow\)

Verification:

Substituting x = -5 in the L.H.S.

\(\Rightarrow\)

L.H.S. = R.H.S.

Hence, verified.

Question 3:

x – 7 = 6

Solution:

x – 7 = 6

Adding 7 on both the sides:

\(\Rightarrow\)

\(\Rightarrow\)

Verification:

Substituting x = 13 in the L.H.S.

\(\Rightarrow\)

L.H.S. = R.H.S.

Hence, verified.

Question 4:

x – 2 = -5

Solution:

x – 2 = -5

Adding 2 on both the sides:

\(\Rightarrow\)

\(\Rightarrow\)

Verification:

Substituting x = -3 in the L.H.S.

\(\Rightarrow\)

L.H.S. = R.H.S.

Hence, verified.

Question 5:

3x – 5 = 13

Solution:

3x – 5 = 13

\(\Rightarrow\)

\(\Rightarrow\)

\(\Rightarrow\)

\(\Rightarrow\)

Verification:

Substituting x = 6 in the L.H.S.

\(\Rightarrow\)

L.H.S. = R.H.S.

Hence, verified.

Question 6:

4x + 7 = 15

Solution:

4x + 7 = 15

\(\Rightarrow\)

\(\Rightarrow\)

\(\Rightarrow\)

\(\Rightarrow\)

Verification:

Substituting x = 2 in the L.H.S.

\(\Rightarrow\)

L.H.S. = R.H.S.

Hence, verified.

Question 7:

\(\frac{x}{5}=12\)

Solution:

\(\frac{x}{5}=12\)

\(\Rightarrow\)

\(\Rightarrow\)

Verification:

Substituting x = 60 in the L.H.S. :

\(\Rightarrow\)

\(\Rightarrow\)

Hence, verified.

Question 8:

\(\frac{3x}{5}=15\)

Solution:

\(\frac{3x}{5}=15\)

\(\Rightarrow\)

\(\Rightarrow\)

\(\Rightarrow\)

\(\Rightarrow\)

Verification:

Substituting x = 25 in the L.H.S. :

\(\Rightarrow\)

\(\Rightarrow\)

Hence, verified.

Question 9:

5x – 3 = x + 17

Solution:

5x – 3 = x + 17

\(\Rightarrow\)

\(\Rightarrow\)

\(\Rightarrow\)

\(\Rightarrow\)

Verification:

Substituting x = 5 on both the sides:

L.H.S. : 5(5) – 3

= 25 – 3

= 22

R.H.S. : 5 + 17 = 22

\(\Rightarrow\)

Hence, verified.

Question 10:

\(2x-\frac{1}{2}=3\)

Solution:

\(2x-\frac{1}{2}=3\)

\(\Rightarrow\)

\(\Rightarrow\)

\(\Rightarrow\)

\(\Rightarrow\)

\(\Rightarrow\)

Verification:

Substituting x = \(\frac{7}{4}\)

\(2\left ( \frac{7}{4} \right )-\frac{1}{2}\)

\(=\frac{7}{2}-\frac{1}{2}=\frac{6}{2}=3=R.H.S.\)

L.H.S. = R.H.S.

Hence, verified.

Question 11:

3(x + 6) = 24

Solution:

3(x + 6) = 24

\(\Rightarrow\)

\(\Rightarrow\)

\(\Rightarrow\)

\(\Rightarrow\)

\(\Rightarrow\)

\(\Rightarrow\)

Verification:

Substituting x = 2 in the L.H.S. :

3(2 + 6) = 3 x 8 = 24 = R.H.S.

L.H.S. = R.H.S.

Hence, verified.

Question 12:

6x + 5 = 2x + 17

Solution:

6x + 5 = 2x + 17

\(\Rightarrow\)

\(\Rightarrow\)

\(\Rightarrow\)

\(\Rightarrow\)

Verification:

Substituting x = 3 on both the sides:

L.H.S. : 6(3) + 5

= 18 + 5

= 23

R.H.S. : 2(3) + 17

= 6 + 17

= 23

L.H.S. = R.H.S.

Hence, verified.

Question 13:

\(\frac{x}{4}-8=1\)

Solution:

\(\frac{x}{4}-8=1\)

\(\Rightarrow\)

\(\Rightarrow\)

\(\Rightarrow\)

or, x = 36

Verification:

Substituting x = 36 in the L.H.S. :

or, \(\frac{36}{4}-8\)

L.H.S. = R.H.S.

Hence, verified.

Question 14:

\(\frac{x}{2}=\frac{x}{3}+1\)

Solution:

\(\frac{x}{2}=\frac{x}{3}+1\)

\(\Rightarrow\)

\(\Rightarrow\)

\(\Rightarrow\)

\(\Rightarrow\)

or, x = 6

Vrification:

Substituting x = 6 on both the sides:

L.H.S. : \(\frac{6}{2}=3\)

R.H.S. : \(\frac{6}{3}+1=2+1=3\)

L.H.S. = R.H.S.

Hence, verified.

Question 15:

3(x + 2) – 2(x – 1) = 7

Solution:

3(x + 2) – 2(x – 1) = 7

\(\Rightarrow\)

or, 3x + 6 – 2x + 2 = 7

or, x + 8 = 7

or, x + 8 – 8 = 7 – 8 [Subtracting 8 from both the sides]

or, x = -1

Verification:

Substituting x = -1 in the L.H.S. :

3(-1 + 2) – 2(-1 – 1)

or, 3(1) – 2(-2)

or, 3 + 4 = 7 = R.H.S.

L.H.S. = R.H.S.

Hence, verified.

Question 16:

5(x – 1) + 2(x + 3) + 6 = 0

Solution:

5(x – 1) + 2(x + 3) + 6 = 0

\(\Rightarrow\)

\(\Rightarrow\)

\(\Rightarrow\)

\(\Rightarrow\)

Verification:

Putting x = -1 in the L.H.S. :

5(-1 – 1) + 2(-1 + 3) + 6

= 5(-2) + 2(2) + 6

= -10 + 5 + 6 = 0 = R.H.S.

Hence, verified.

Question 17:

6(1 – 4x) + 7(2 + 5x) = 53

Solution:

6(1 – 4x) + 7(2 + 5x) = 53

or, \(6\times 1-6\times 4x+7\times 2+7\times 5x=53\)

or, 6 – 24x + 14 + 35x = 53

or, 11x + 20 = 53

or, 11x + 20 – 20 = 53 – 20 [Subtracting 20 from both the sides]

or, 11x = 33

or, \(\frac{11x}{11}=\frac{33}{11}\)

or, x = 3

Verification:

Substituting x = 3 in the L.H.S. :

6(1 – 4 x 3) + 7(2 + 5 x 3)

= 6(1 – 12) + 7(2 + 15)

= 6(-11) + 7(17)

= -66 + 119 = 53 = R.H.S.

L.H.S. = R.H.S.

Hence, verified.

Question 18:

16(3x – 5) – 10(4x – 8) = 40

Solution:

16(3x – 5) – 10(4x – 8) = 40

or, \(16\times 3x-16\times 5-10\times 4x-10\times (-8)=40\)

or, 48x – 80 – 40x + 80 = 40

or, 8x = 40

or, \(\frac{8x}{8}=\frac{40}{8}\)

or, x = 5

Verification:

Substituting x = 5 in the L.H.S. :

16(3 x 5 – 5) – 10(4 x 5 – 8)

= 16(15 – 5) – 10(20 – 8)

= 16(10) – 10(12)

= 160 – 120 = 40 = R.H.S.

L.H.S. = R.H.S.

Hence, verified.

Question 19:

3(x + 6) + 2(x + 3) = 64

Solution:

3(x + 6) + 2(x + 3) = 64

\(\Rightarrow\)

\(\Rightarrow\)

\(\Rightarrow\)

\(\Rightarrow\)

\(\Rightarrow\)

\(\Rightarrow\)

\(\Rightarrow\)

Verification:

Substituting x = 8 in the L.H.S. :

3(8 + 6) + 2(8 + 3)

= 3(14) + 2(11)

= 42 + 22 = 64 = R.H.S.

L.H.S. = R.H.S.

Hence, verified.

Question 20:

3(2 – 5x) – 2(1 – 6x) = 1

Solution:

3(2 – 5x) – 2(1 – 6x) = 1

or, \(3\times 2+3\times (-5x)-2\times 1-2\times (-6x)=1\)

or, 6 – 15x – 2 + 12x = 1

or, 4 – 3x = 1

or, 3 = 3x

or, x = 1

Verification:

Substituting x = 1 in the L.H.S. :

3(2 – 5 x 1) – 2(1 – 6 x 1)

= 3(2 – 5) – 2(1 – 6)

= 3(-3) – 2(-5)

= -9 + 10 = 1 = R.H.S.

L.H.S. = R.H.S.

Hence, verified.

Question 21:

\(\frac{n}{4}-5=\frac{n}{6}+\frac{1}{2}\)

Solution:

\(\frac{n}{4}-5=\frac{n}{6}+\frac{1}{2}\)

or, \(\frac{n}{4}-\frac{n}{6}=\frac{1}{2}+5\)

or,\(\frac{3n-2n}{12}=\frac{1+10}{2}\)

or, \(\frac{n}{12}=\frac{11}{2}\)

or, \(\frac{n}{12}\times 12=\frac{11}{2}\times 12\)

or, n = 66

Verification:

Substituting n = 66 on both the sides:

L.H.S. :

\(\frac{66}{4}-5=\frac{33}{2}=5=\frac{33-10}{2}=\frac{23}{2}\)

R.H.S. :

\(\frac{66}{6}+\frac{1}{2}=11+\frac{1}{2}=\frac{22+1}{2}=\frac{23}{2}\)

L.H.S. = R.H.S.

Hence, verified.

Question 22:

\(\frac{2m}{3}+8=\frac{m}{2}-1\)

Solution:

\(\frac{2m}{3}+8=\frac{m}{2}-1\)

or, \(\frac{2m}{3}-\frac{m}{2}=-1-8\)

or, \(\frac{4m-3m}{6}=-9\)

or, \(\frac{m}{6}=-9\)

or, \(\frac{m}{6}\times 6=-9\times 6\)

or, m = -54

Verification:

Substituting x = -54 on both the sides:

L.H.S. :

\(\frac{2(-54)}{3}+8=\frac{-54}{2}-1\)

= \(\frac{-108}{3}+8\)

= -36 + 8

= -28

R.H.S. :

\(\frac{-54}{2}-1 \)

= -27 – 1

= -28

L.H.S. = R.H.S.

Hence, verified

Question 23:

\(\frac{2x}{5}-\frac{3}{2}=\frac{x}{2}+1\)

Solution:

\(\frac{2x}{5}-\frac{3}{2}=\frac{x}{2}+1\)

or,\(\frac{2x}{5}-\frac{x}{2}=1+\frac{3}{2}\)

or,\(\frac{4x-5x}{10}=\frac{2+3}{2}\)

or, \(\frac{-x}{10}=\frac{5}{2}\)

or, \(\frac{-x}{10}\left ( -10 \right )=\frac{5}{2}\times \left ( -10 \right )\)

or, x = -25

Verification:

Substituting x = -25 on both the sides:

L.H.S. :\(\frac{2(-25)}{5}-\frac{3}{2}\)

= \(\frac{-50}{5}-\frac{3}{2}\)

= \(-10-\frac{3}{2}=\frac{-23}{2}\)

R.H.S. : \(\frac{-25}{2}+1=\frac{-25+2}{2}=\frac{-23}{2}\)

L.H.S. = R.H.S.

Hence, verified.

Question 24:

\(\frac{x-3}{5}-2=\frac{2x}{5}\)

Solution:

\(\frac{x-3}{5}-2=\frac{2x}{5}\)

or, \(\frac{x}{5}-\frac{3}{5}-2=\frac{2x}{5}\)

or, \(-\frac{3}{5}-2=\frac{2x}{5}-\frac{x}{5}\)

or, \(\frac{-3-10}{5}=\frac{x}{5}\)

or, \(\frac{-13}{5}=\frac{x}{5}\)

or, \(\frac{-13}{5}(5)=\frac{x}{5}\times (5)\)

or, x = -13

Verification:

Substituting x = -13 on both the sides:

L.H.S. : \(\frac{-13-3}{5}-2\)

= \(\frac{-16}{5}-2=\frac{-16-10}{5}=\frac{-26}{5}\)

R.H.S. : \(\frac{2\times (-13)}{5}=\frac{-26}{5}\)

L.H.S. = R.H.S.

Hence, verified.

Question 25:

\(\frac{3x}{10}-4=14\)

Solution:

\(\frac{3x}{10}-4=14\)

or, \(\frac{3x}{10}-4+4=14+4\)

or, \(\frac{3x}{10}18\)

or, \(\frac{3x}{10}\times 10=18\times 10\)

or, 3x = 180

or, \(\frac{3x}{3}=\frac{180}{3}\)

or, x = 60

Verification:

Substituting x = 60 on the L.H.S. :

\(\frac{3\times 60}{10}-4\)

= \(\frac{180}{10}-4=18-4=14=R.H.S.\)

L.H.S. = R.H.S.

Hence, verified.

Question 26:

\(\frac{3}{4}(x-1)=x-3\)

Solution:

\(\frac{3}{4}(x-1)=x-3\)

\(\Rightarrow\)

\(\Rightarrow\)

\(\Rightarrow\)

\(\Rightarrow\)

\(\Rightarrow\)

\(\Rightarrow\)

or, x = 9

Verification:

Substituting x = 9 on both the sides:

L.H.S. : \(\frac{3}{4}(9-1)\)

=\(\frac{3}{4}(8)\)

= 6

R.H.S. : 9 – 3 = 6

L.H.S. = R.H.S.

Hence, verified.