**Solve the following equations and verify your answer:**

**Question 1. (2x-3)/(3x+2) = -2/3**

**Solution:**

Given:

(2x-3) / (3x+2) = -2/3

After cross-multiplication we will get,

3(2x – 3) = -2(3x + 2)

6x – 9 = -6x – 4

Now rearrange the above equation

6x + 6x = 9 – 4

12x = 5

x = 5/12

Now verify the given equation by putting x = 5/12

(2x-3) / (3x+2) = -2/3

x = 5/12

(2(5/12) – 3) / (3(5/12) + 2) = -2/3

((5/6)-3) / ((5/4) + 2) = -2/3

((5-18)/6) / ((5+8)/4) = -2/3

(-13/6) / (13/4) = -2/3

(-13/6) × (4/13) = -2/3

-4/6 = -2/3

-2/3 = -2/3

Here, L.H.S. = R.H.S.,

Thus, the given equation is verified.

**Question 2. (2-y)/(y+7) = 3/5**

**Solution:**

Given:

(2-y) / (y+7) = 3/5

After cross-multiplication we will get,

5(2-y) = 3(y+7)

10 – 5y = 3y + 21

Now rearrange the above equation,

10 – 21 = 3y + 5y

8y = – 11

y = -11/8

Now verify the given equation, by putting y = -11/8

(2-y) / (y+7) = 3/5

y = -11/8

(2 – (-11/8)) / ((-11/8) + 7) = 3/5

((16+11)/8) / ((-11+56)/8) = 3/5

(27/8) / (45/8) = 3/5

(27/8) × (8/45) = 3/5

27/45 = 3/5

3/5 = 3/5

Here, L.H.S. = R.H.S.,

Thus, the given equation is verified.

**Question 3. (5x – 7)/(3x) = 2**

**Solution:**

Given:

(5x – 7) / (3x) = 2

After cross-multiplication we will get,

5x – 7 = 2(3x)

5x – 7 = 6x

5x – 6x = 7

-x = 7

x = -7

Now verify the given equation but putting x = -7

(5x – 7) / (3x) = 2

x = -7

(5(-7) – 7) / (3(-7)) = 2

(-35 – 7) / -21 = 2

-42/-21 = 2

2 = 2

Here, L.H.S. = R.H.S.,

Thus, the given equation is verified.

**Question 4. (3x+5)/(2x + 7) = 4**

**Solution:**

Given:

(3x+5) / (2x + 7) = 4

After cross-multiplication we will get,

3x + 5 = 4(2x+7)

3x + 5 = 8x + 28

3x – 8x = 28 – 5

-5x = 23

x = -23/5

Now verify the given equation by putting x =-23/5

(3x+5) / (2x + 7) = 4

x = -23/5

(3(-23/5) + 5) / (2(-23/5) + 7) = 4

(-69/5 + 5) / (-46/5 + 7) = 4

(-69+25)/5 / (-46+35)/5 = 4

-44/5 / -11/5 = 4

-44/5 × 5/-11 = 4

44/11 = 4

4 = 4

Here, L.H.S. = R.H.S.,

Thus, the given equation is verified.

**Question 5. (2y + 5)/(y + 4) = 1**

**Solution:**

Given:

(2y + 5) / (y + 4) = 1

After cross-multiplication we will get,

2y + 5 = y + 4

2y – y = 4 – 5

y = -1

Now verify the given equation by substituting y = -1

(2y + 5) / (y + 4) = 1

y = -1

(2(-1) + 5) / (-1 + 4) = 1

(-2+5) / 3 = 1

3/3 = 1

1 = 1

Here, L.H.S. = R.H.S.,

Thus, the given equation is verified.

**Question 6. (2x + 1)/(3x – 2) = 5/9**

**Solution:**

Given:

(2x + 1) / (3x – 2) = 5/9

After cross-multiplication we will get,

9(2x + 1) = 5(3x – 2)

18x + 9 = 15x – 10

18x – 15x = -10 – 9

3x = -19

x = -19/3

Now verify the given equation byt substituting x = -19/3

(2x + 1) / (3x – 2) = 5/9

x = -19/3

(2(-19/3) + 1) / (3(-19/3) – 2) = 5/9

(-38/3 + 1) / (-57/3 – 2) = 5/9

(-38 + 3)/3 / (-57 – 6)/3 = 5/9

-35/3 / -63/3 = 5/9

-35/3 × 3/-63 = 5/9

-35/-63 = 5/9

5/9 = 5/9

Here, L.H.S. = R.H.S.,

Thus, the given equation is verified.

**Question 7. (1 – 9y)/(19 – 3y) = 5/8**

**Solution:**

Given:

(1 – 9y) / (19 – 3y) = 5/8

After cross-multiplication we willget,

8(1- 9y) = 5(19-3y)

8 – 72y = 95 – 15y

8 – 95 = 72y – 15y

57y = -87

y = -87/57

y = -29/19

Now verify the given equation by substituting y = -29/19

(1 – 9y) / (19 – 3y) = 5/8

y = -29/19

(1 – 9(-29/19)) / (19 – 3(-29/19)) = 5/8

(19+261)/19 / (361+87)/19 = 5/8

280/19 × 19/448 = 5/8

280/ 448 = 5/8

5/8 = 5/8

Here, L.H.S. = R.H.S.,

Thus, the given equation is verified.

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

**Solution:**

Given:

2x / (3x + 1) = 1

After cross-multiplication we will get,

2x = 1(3x + 1)

2x = 3x + 1

2x – 3x = 1

-x = 1

x = -1

Now verify the given equation by substituting x = -1

2x / (3x + 1) = 1

x = -1

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

-2 /(-3+1) = 1

-2/-2 = 1

1 = 1

Here, L.H.S. = R.H.S.,

Thus the given equation is verified.

**Question 9. y – (7 – 8y)/9y – (3 + 4y) = 2/3**

**Solution:**

Given:

y – (7 – 8y)/9y – (3 + 4y) = 2/3

(y – 7 + 8y) / (9y – 3 – 4y) = 2/3

(-7 + 9y) / (5y – 3) = 2/3

After cross-multiplication we will get,

3(-7 + 9y) = 2(5y – 3)

-21 + 27y = 10y – 6

27y – 10y = 21 – 6

17y = 15

y = 15/17

Now verify the given equation by substituting y = 15/17

y – (7 – 8y)/9y – (3 + 4y) = 2/3

y = 15/17

15/17 – (7-8(15/17))/ 9(15/17) – (3 + 4(15/17)) = 2/3

15/17 – (7 – 120/17) / 135/17 – (3 + 60/17) = 2/3

15/17 – ((119-120)/17) / 135/17 – ((51+60)/17) = 2/3

15/17 – (-1/17) / 135/17 – (111/17) = 2/23

((15 + 1)/17) / ((135-111)/17) = 2/3

16/17 / 24/17 = 2/3

16/24 = 2/3

2/3 = 2/3

Here, L.H.S. = R.H.S.,

Thus, the given equation is verified.

**Question 10. 6/2x – (3 – 4x) = 2/3**

**Solution:**

Given:

6/ 2x – (3 – 4x) = 2/3

6/(2x – 3 + 4x) = 2/3

6/(6x – 3) = 2/3

After cross-multiplication we will get,

3(6) = 2(6x – 3)

18 = 12x – 6

12x = 18 + 6

12x = 24

x = 24/12

x = 2

Now verify the given equation by substituting x = 2

6/ 2x – (3 – 4x) = 2/3

6/(6x – 3) = 2/3

x = 2

6/ (6(2) – 3) = 2/3

6/(12-3) = 2/3

6/9 = 2/3

2/3 = 2/3

Here, L.H.S. = R.H.S.,

Thus the given equation is verified.

**Question 11. 2/3x – 3/2x = 1/12**

**Solution:**

Given:

2/3x – 3/2x = 1/12

By taking LCM for 2 and 3 which is 6

4-9/6x = 1/12

-5/6x = 1/12

After cross-multiplying we will get,

12(-5) = 1 (6x)

-60 = 6x

x = -60/6

x = -10

Now verify the given equation by substituting x = -10

2/3x – 3/2x = 1/12

x = -10

2/3(-10) – 3/2(-10) = 1/12

2/-30 – 3/-20 = 1/12

-4+6/60 = 1/12

5/60 = 1/12

1/12 = 1/12

Here, L.H.S. = R.H.S.,

Thus, the given equation is verified.

**Question 12. (3x + 5)/(4x + 2) = (3x + 4)/(4x + 7)**

**Solution:**

Given:

(3x + 5)/ (4x + 2) = (3x + 4)/(4x + 7)

(3x + 5)/ (4x + 2) – (3x + 4)/(4x + 7) = 0

By taking LCM as (4x + 2) (4x + 7)

((3x + 5) (4x + 7) – (3x + 4) (4x + 2)) / (4x + 2) (4x + 7) = 0

After cross-multiplying we will get,

(3x + 5) (4x + 7) – (3x + 4) (4x + 2) = 0

(3x + 5) (4x + 7) – (3x + 4) (4x + 2) = 0

12x

^{2}+ 21x + 20x + 35 – 12x^{2}– 6x – 16x – 8 = 019x + 35 – 8 = 0

19x = -27

x = -27/19

Now verify the given equation by substituting, x = -27/19

(3x + 5)/ (4x + 2) = (3x + 4)/(4x + 7)

x = -27/19

(3(-27/19) +5) / (4(-27/19) + 2) = (3(-27/19) + 4) / (4(-27/19) + 7)

(-81/19 + 5) / (-108/19 + 2) = (-81/19 + 4) / (-108/19 + 7)

((-81+95)/19) / ((-108+38)/19) = ((-81+76)/19) / ((-108+133)/19)

14/19 / -70/19 = -5/19 / 25/19

-14/70 = -5/25

-1/5 = -1/5

Here, L.H.S. = R.H.S.,

Thus, the given equation is verified.