**Solve the following linear equations.**

**1. x/2 – 1/5 = x/3 + ¼**

**Solution:**

Given, x/2 – 1/5 = x/3 + ¼

Now,

x/2 – x/3 = ¼+ 1/5

(3x – 2x)/6 = (5 + 4)/20

3x – 2x = 9/20 × 6

x = 54/20

x = 27/10

**2. n/2 – 3n/4 + 5n/6 = 21**

**Solution:**

Given, n/2 – 3n/4 + 5n/6 = 21

Now,

(6n – 9n + 10n)/12 = 21

7n/12 = 21

7n = 21 × 12

n = 252/7

n = 36

**3. x + 7 – 8x/3 = 17/6 – 5x/2**

**Solution:**

Given, x + 7 – 8x/3 = 17/6 – 5x/2

Now,

x – 8x/3 + 5x/2 = 17/6 – 7

(6x – 16x + 15x)/6 = (17 – 42)/6

5x/6 = – 25/6

5x = – 25

x = – 5

**4. (x – 5)/3 = (x – 3)/5**

**Solution:**

Given, (x – 5)/3 = (x – 3)/5

Now,

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

5x-25 = 3x-9

5x – 3x = -9+25

2x = 16

x = 8

**5. (3t – 2)/4 – (2t + 3)/3 = 2/3 – t**

**Solution:**

Given, (3t – 2)/4 – (2t + 3)/3 = 2/3 – t

Now,

((3t – 2)/4) × 12 – ((2t + 3)/3) × 12

(3t – 2) × 3 – (2t + 3) × 4 = 2 × 4 – 12t

9t – 6 – 8t – 12 = 8 – 12t

9t – 6 – 8t – 12 = 8 – 12t

t – 18 = 8 – 12t

t + 12t = 8 + 18

13t = 26

t = 2

**6. m – (m – 1)/2 = 1 – (m – 2)/3**

**Solution:**

Given, m – (m – 1)/2 = 1 – (m – 2)/3

Now,

m – m/2 – 1/2 = 1 – (m/3 – 2/3)

m – m/2 + ½ = 1 – m/3 + 2/3

m – m/2 + m/3 = 1 + 2/3 – ½

m/2 + m/3 = ½ + 2/3

(3m + 2m)/6 = (3 + 4)/6

5m/6 = 7/6

m = 7/6 × 6/5

m = 7/5

**Simplify and solve the following linear equations.**

**7. 3(t – 3) = 5(2t + 1)**

**Solution:**

Given, 3(t – 3) = 5(2t + 1)

Now,

3t – 9 = 10t + 5

3t – 10t = 5 + 9

-7t = 14

t = 14/-7

t = -2

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

**Solution**:

Given, 15(y – 4) –2(y – 9) + 5(y + 6) = 0

Now,

15y – 60 -2y + 18 + 5y + 30 = 0

15y – 2y + 5y = 60 – 18 – 30

18y = 12

y = 12/18

y = 2/3

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

**Solution:**

Given, 3(5z – 7) – 2(9z – 11) = 4(8z – 13) – 17

Now,

15z – 21 – 18z + 22 = 32z – 52 – 17

15z – 18z – 32z = -52 – 17 + 21 – 22

-35z = -70

z = -70/-35

z = 2

**10. 0.25(4f – 3) = 0.05(10f – 9)**

**Solution:**

Given, 0.25(4f – 3) = 0.05(10f – 9)

Now,

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

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