#### Solve simultaneous linear equations:

(i) x/6 = y – 6

3x/4 = 1 + y

(ii) x – 2/3 y = 8/3

2x/5 – y = 7/5.

**Solution:**

(i) x/6 = y – 6

3x/4 = 1 + y

We can write it as

x = 6 (y – 6)

x = 6y – 36

x – 6y = – 36 ….. (1)

3x/4 = 1 + y

By cross multiplication

3x = 4 (1 + y)

So we get

3x = 4 + 4y

3x – 4y = 4 …. (2)

Multiply equation (1) by 3

3x – 18y = – 108

3x – 4y = 4

Subtracting both the equations

– 14y = – 112

So we get

y = – 112/- 14 = 8

Substitute the value of y in equation (1)

x – 6 × 8 = – 36

By further calculation

x – 48 = – 36

x = – 36 + 48

x = 12

Therefore, x = 12 and y = 8.

(ii) x – 2/3 y = 8/3

2x/5 – y = 7/5

We can write it as

x – 2/3 y = 8/3

Taking LCM

(3x – 2y)/ 3 = 8/3

By cross multiplication

3x – 2y = 8/3 × 3 = 8

3x – 2y = 8 ….. (1)

2x/5 – y = 7/5

Taking LCM

(2x – 5y)/ 5 = 7/5

By cross multiplication

2x – 5y = 7/5 × 5 = 7

2x – 5y = 7 …… (2)

Multiply equation (1) by 2 and (2) by 3

6x – 4y = 16 ….. (3)

6x – 15y = 21 …… (4)

Subtracting both the equations

11y = – 5

y = – 5/11

Substitute the value of y in equation (1)

3x – 2 (-5/11) = 8

By further calculation

3x + 10/11 = 8

We can write it as

3x = 8 – 10/11

Taking LCM

3x = (88 – 10)/ 11 = 78/11

By cross multiplication

x = 78/ (11 × 3) = 26/11

Therefore, x = 26/11 and y = – 5/11.

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