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Solve the following simultaneous equations.

\({7x -2y \over xy} = 5\)  \({{8x + 7y} \over xy} =15 \)

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Sol. 

Consider, \({7x -2y \over xy} = 5\)

\({7x \over xy } - {2y \over xy } = 5 \)

\({-2 \over x} +{ 7 \over y} =5\)

\(-2({1 \over x}) + 7 ({1 \over y}) = 5 \)   ...(I) 

Now consider 

\({{8x + 7y} \over xy }=15\)

∴  \({8x \over xy} + {7y \over xy}=15\)

∴ \({8 \over y} + {7 \over x} = 15\)

∴ \({7 \over x} + {8 \over y} = 15\)

∴ \(7 ({1 \over x}) + 8({1 \over y})= 15 \)  ...(II)

Replace  \(1 \over x \)  by a  and  \(1 \over y\) by b in equations (I) and (II), we get

–2a + 7b = 5 ...(III)

7a + 8b = 15 ...(IV) 

Multiplying eq. (III) by 7 and eq. (IV) by 2 

–14a + 49b = 35 ...(V)

14a + 16b  = 30 ...(VI)

Adding eq. (V) and (VI)

–14a + 49 b = 35

+ 14 a + 16 b = 30

               65b = 65

                   b =1 

Place b = 1 in eq. (III)

–2a + 7(1) = 5
∴  –2a + 7 = 5

∴ –2a = 5 – 7

 ∴ –2a = –2

∴ a =1

Resubstituting the values of a and b 

\({1 \over x} = 1\)  and  \({1 \over y}=1\)

∴ x =  1         ∴ y =  1

∴ (x, y) = (1, 1) is the required solutions.
 

Linear equation in two variables August 04 , 2018 0 Comments 53 views