**Solve each of the following equations and also check your results in each case:**

**Question 1. (2x + 5)/3 = 3x – 10**

**Solution:**

First simplify the equation,

(2x + 5)/3 – 3x = – 10

By taking LCM

(2x + 5 – 9x)/3 = -10

(-7x + 5)/3 = -10

After cross-multiplication we get,

-7x + 5 = -30

-7x = -30 – 5

-7x = -35

x = -35/-7

= 5

Now verify the equation by putting x = 5.

(2x + 5)/3 = 3x – 10

x = 5

(2×5 + 5)/3 = 3(5) – 10

(10+5)/3 = 15-10

15/3 = 5

5 = 5

Thus, L.H.S. = R.H.S.,

Hence, the equation is verified.

**Question 2. (a – 8)/3 = (a – 3)/2**

**Solution:**

After cross-multiplication we will get,

(a – 8)2 = (a-3)3

2a – 16 = 3a – 9

2a – 3a = -9 + 16

-a = 7

a = -7

Now verify the equation by putting a = -7.

(a – 8)/3 = (a – 3)/2

a = -7

(-7 – 8)/3 = (-7 – 3)/2

-15/3 = -10/2

-5 = -5

Thus, L.H.S. = R.H.S.,

Hence, the equation is verified.

**Question 3. (7y + 2)/5 = (6y – 5)/11**

**Solution:**

After cross-multiplication we will get,

(7y + 2)11 = (6y – 5)5

77y + 22 = 30y – 25

77y – 30y = -25 – 22

47y = -47

y = -47/47

y = -1

Now verify the equation by putting y = -1.

(7y + 2)/5 = (6y – 5)/11

y =-1

(7(-1) + 2)/5 = (6(-1) – 5)/11

(-7 + 2)/5 = (-6 – 5)/11

-5/5 = -11/11

-1 = -1

Thus, L.H.S. = R.H.S.,

Hence, the equation is verified.

**Question 4. x – 2x + 2 – 16/3x + 5 = 3 – 7/2x**

**Solution:**

x – 2x + 2 – 16/3x + 5 = 3 – 7/2x

First rearrange the equation

x – 2x – 16x/3 + 7x/2 = 3 – 2 – 5

By taking LCM for 2 and 3 which is 6

(6x – 12x – 32x + 21x)/6 = -4

-17x/6 = -4

After cross-multiplying we will get,

-17x = -4×6

-17x = -24

x = -24/-17

x = 24/17

Now verify the equation by putting x = 24/17.

x – 2x + 2 – 16/3x + 5 = 3 – 7/2x

x = 24/17

24/17 – 2(24/17) + 2 – (16/3)(24/17) + 5 = 3 – (7/2)(24/17)

24/17 – 48/17 + 2 – 384/51 + 5 = 3 – 168/34

By taking 51 and 17 as the LCM we get,

(72 – 144 + 102 – 384 + 255)/51 = (102 – 168)/34

-99/51 = -66/34

-33/17 = -33/17

Thus, L.H.S. = R.H.S.,

Hence, the equation is verified.

**Question 5. 1/2x + 7x – 6 = 7x + 1/4**

**Solution:**

1/2x + 7x – 6 = 7x + 1/4

First rearrange the equation

1/2x + 7x – 7x = 1/4 + 6 (by taking LCM)

1/2x = (1+ 24)/4

1/2x = 25/4

After cross-multiplying

4x = 25 × 2

4x = 50

x = 50/4

x = 25/2

Now verify the equation by putting x = 25/2.

1/2x + 7x – 6 = 7x + 1/4

x = 25/2

(1/2) (25/2) + 7(25/2) – 6 = 7(25/2) + 1/4

25/4 + 175/2 – 6 = 175/2 + 1/4

By taking LCM for 4 and 2 is 4

(25 + 350 – 24)/4 = (350+1)/4

351/4 = 351/4

Thus, L.H.S. = R.H.S.,

Hence, the equation is verified.

**Question 6. 3/4x + 4x = 7/8 + 6x – 6**

**Solution:**

3/4x + 4x = 7/8 + 6x – 6

First rearrange the equation

3/4x + 4x – 6x = 7/8 – 6

By taking 4 and 8 as LCM

(3x + 16x – 24x)/4 = (7 – 48)/8

-5x/4 = -41/8

After cross-multiplying

-5x(8) = -41(4)

-40x = -164

x = -164/-40

= 82/20

= 41/10

Now verify the equation by putting x = 41/10.

3/4x + 4x = 7/8 + 6x – 6

x = 41/10

(3/4)(41/10) + 4(41/10) = 7/8 + 6(41/10) – 6

123/40 + 164/10 = 7/8 + 246/10 – 6

(123 + 656)/40 = (70 + 1968 – 480)/80

779/40 = 1558/80

779/40 = 779/40

Thus, L.H.S. = R.H.S.,

Hence, the equation is verified.

**Question 7. 7x/2 – 5x/2 = 20x/3 + 10**

**Solution:**

7x/2 – 5x/2 = 20x/3 + 10

First rearrange the equation

7x/2 – 5x/2 – 20x/3 = 10

By taking LCM for 2 and 3 is 6

(21x – 15x – 40x)/6 = 10

-34x/6 = 10

After cross-multiplying

-34x = 60

x = 60/-34

= -30/17

Now verify the equation by putting x = -30/7.

7x/2 – 5x/2 = 20x/3 + 10

x = -30/7

(7-/2)(-30/17) – (5/2)(-30/17) = (20/3)(-30/17) + 10

-210/34 +150/34 = -600/51 + 10

-30/17 = (-600+510)/51

= -90/51

-30/17 = -30/17

Thus, L.H.S. = R.H.S.,

Hence, the equation is verified.

**Question 8. (6x + 1)/2 + 1 = (7x – 3)/3**

**Solution:**

(6x+1)/2 + 1 = (7x-3)/3

(6x + 1 + 2)/2 = (7x – 3)/3

After cross-multiplying

(6x + 3)3 = (7x – 3)2

18x + 9 = 14x – 6

18x – 14x = -6 – 9

4x = -15

x = -15/4

Now verify the equation by putting x = -15/4.

(6x+1)/2 + 1 = (7x-3)/3

x = -15/4

(6(-15/4) + 1)/2 + 1 = (7(-15/4) – 3)/3

(3(-15/2) + 1)/2 + 1 = (-105/4 -3)/3

(-45/2 + 1)/2 + 1 = (-117/4)/3

(-43/4) + 1 = -117/12

(-43+4)/4 = -39/4

-39/4 = -39/4

Thus, L.H.S. = R.H.S.,

Hence, the equation is verified.

**Question 9. (3a-2)/3 + (2a+3)/2 = a + 7/6**

**Solution:**

(3a-2)/3 + (2a+3)/2 = a + 7/6

First rearrange the equation

(3a-2)/3 + (2a+3)/2 – a = 7/6

By taking LCM for 2 and 3 which is 6

((3a-2)2 + (2a+3)3 – 6a)/6 = 7/6

(6a – 4 + 6a + 9 – 6a)/6 = 7/6

(6a + 5)/6 = 7/6

6a + 5 = 7

6a = 7-5

6a = 2

a = 2/6

a = 1/3

Now verify the equation by putting a = 1/3.

(3a-2)/3 + (2a+3)/2 = a + 7/6

a = 1/3

(3(1/3)-2)/3 + (2(1/3) + 3)/2 = 1/3 + 7/6

(1-2)/3 + (2/3 + 3)/2 = (2+7)/6

-1/3 + (11/3)/2 = 9/6

-1/3 + 11/6 = 3/2

(-2+11)/6 = 3/2

9/6 = 3/2

3/2 = 3/2

Thus, L.H.S. = R.H.S.,

Hence, the equation is verified.

**Question 10. x – (x – 1)/2 = 1 – (x – 2)/3**

**Solution:**

x – (x-1)/2 = 1 – (x-2)/3

First rearrange the equation

x – (x-1)/2 + (x-2)/3 = 1

By taking LCM for 2 and 3 which is 6

(6x – (x-1)3 + (x-2)2)/6 = 1

(6x – 3x + 3 + 2x – 4)/6 = 1

(5x – 1)/6 = 1

After cross-multiplying

5x – 1 = 6

5x = 6 + 1

x = 7/5

Now verify the equation by putting x = 7/5.

x – (x-1)/2 = 1 – (x-2)/3

x = 7/5

7/5 – (7/5 – 1)/2 = 1 – (7/5 – 2)/3

7/5 – (2/5)/2 = 1 – (-3/5)/3

7/5 – 2/10 = 1 + 3/15

(14 – 2)/10 = (15+3)/15

12/10 = 18/15

6/5 = 6/5

Thus, L.H.S. = R.H.S.,

Hence, the equation is verified.

**Question 11. 3x/4 – (x-1)/2 = (x-2)/3**

**Solution:**

3x/4 – (x-1)/2 = (x-2)/3

First rearrange the equation

3x/4 – (x-1)/2 – (x-2)/3 = 0

By taking LCM for 4, 2 and 3 which is 12

(9x – (x-1)6 – (x-2)4)/12 = 0

(9x – 6x + 6 – 4x + 8)/12 = 0

(-x + 14)/12 = 0

After cross-multiplying

-x + 14 = 0

x = 14

Now verify the equation by putting x = 14

3x/4 – (x-1)/2 = (x-2)/3

x = 14

3(14)/4 – (14-1)/2 = (14-2)/3

42/4 – 13/2 = 12/3

(42 – 26)/4 = 4

16/4 = 4

4 = 4

Thus, L.H.S. = R.H.S.,

Hence, the equation is verified.

**Question 12. 5x/3 – (x-1)/4 = (x-3)/5**

**Solution:**

5x/3 – (x-1)/4 = (x-3)/5

First rearrange the equation

5x/3 – (x-1)/4 – (x-3)/5 = 0

By taking LCM for 3, 4 and 5 which is 60

((5x × 20) – (x-1)15 – (x-3)12)/60 = 0

(100x – 15x + 15 -12x + 36)/60 = 0

(73x + 51)/60 = 0

After cross-multiplying

73x + 51 = 0

x = -51/73

Now verify the equation by putting x = -51/73

5x/3 – (x-1)/4 = (x-3)/5

x = -51/73

(20x – (x-1)3)/12 = (-51/73 – 3)/5

(20x – 3x + 3)/12 = (-270/73)/5

(17x + 3)/12 = -270/365

(17(-51/73) + 3)/12 = -54/73

(-867/73 + 3)/12 = -54/73

((-867 + 219)/73)/12 = -54/73

(-648)/876 = -54/73

-54/73 = -54/73

Thus, L.H.S. = R.H.S.,

Hence, the equation is verified.

**Question 13. (3x+1)/16 + (2x-3)/7 = (x+3)/8 + (3x-1)/14**

**Solution:**

(3x+1)/16 + (2x-3)/7 = (x+3)/8 + (3x-1)/14

First rearrange the equation

(3x+1)/16 + (2x-3)/7 – (x+3)/8 – (3x-1)/14 = 0

By taking LCM for 16, 7, 8 and 14 which is 112

((3x+1)7 + (2x-3)16 – (x+3)14 – (3x-1)8)/112 = 0

(21x + 7 + 32x – 48 – 14x – 42 – 24x + 8)/112 = 0

(21x + 32x – 14x – 24x + 7 – 48 – 42 + 8)/112 = 0

(15x – 75)/112 = 0

After cross-multiplying

15x – 75 = 0

15x = 75

x = 75/15

x = 5

Now verify the equation by putting x = 5

(3x+1)/16 + (2x-3)/7 = (x+3)/8 + (3x-1)/14

x = 5

(3(5)+1)/16 + (2(5)-3)/7 = (5+3)/8 + (3(5)-1)/14

(15+1)/16 + (10-3)/7 = 8/8 + (15-1)/14

16/16 + 7/7 = 8/8 + 14/14

1 + 1 = 1 + 1

2 = 2

Thus, L.H.S. = R.H.S.,

Hence, the equation is verified.