Linear Equation 22

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Solve : 7 / x + 8 / y = 2 and 2 / x + 12 / y = 20

Solution:

Given, 7 / x + 8 / y = 2 ----------------------- (i)

and, 2 / x + 12 / y = 20 ----------------------- (ii)

Multiplying equation (i) by 2 and equation (ii) by 7, we get

14 / x + 16 / y = 4 --------------------------- (iv)

14 / x + 84 / y = 140 --------------------------- (v)

Subtracting (v) from (iv), we get

- 68 / y = - 136

=> - 136 y = - 68 ; or y = 1 / 2

Substituting y = 1 / 2 in equation (i), we get

7 / x + 8 X 2 / 1 = 2

=> 7 / x = 2 - 16

=> x = - 1 /2

Therefore, x = - 1 / 2 and y = 1 / 2

Alternative Method:

Let 1 / x = a and 1 / y = b

Therefore, 7 / x + 8 / y = 2 becomes : 7 a + 8 b = 2 -------- (i)

and, 2 / x + 12 / y = 20 becomes : 2 a + 12 b = 20 ---------- (ii)

Multiplying equation (i) by 2 and equation (ii) by 7 we get

14 a + 16 b = 4 --------------------------------------------- (iii)

14 a + 84 b = 140 ------------------------------------------- (iv)

Subtracting (iv) from (iii) , we get

- 68 b = - 136 ; => b = (- 136) / ( - 68) = 2

i.e. 1 / y = 2 and y = 1 / 2

Now, 2 a + 12 b = 20

=> 2 a + 12 X 2 = 20

=> 2 a + 24 = 20

=> 2 a = - 4 and a = - 2

i.e. 1 / x = - 2 and x = - 1 / 2

Therefore, Solution is : x = - 1 / 2 and y = 1 / 2

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