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Question
Solve each of the following systems of equations by the method of cross-multiplication :
`57/(x + y) + 6/(x - y) = 5`
`38/(x + y) + 21/(x - y) = 9`
Solution
Let `1/(x +y) = u and 1/(x -y ) = v` Then the given system of equations become
`57u + 6v = 5 => 57u + 6v - 5 = 0`
``38u + 21v = 9 => 38u + 21v - 9 = 0`
Here
`a_1 = 57, b_1 = 6, c_1 = -5`
`a_2 = 38, b_2 = 21, c_2 = -9`
By cross multiplication, we have
`=> u/(-54 + 105) = (-v)/(-513 + 190) = 1/(1193 - 228)`
`=> u/51 = (-v)/(-323) = 1/969`
`=> u/51 = v/323 = 1/969`
Now
`u/51= 1/969`
`=> u = 51/969`
`=> u = 1/19`
And
`v/(323) = 1/969`
`=> v = 323/969`
`=> v = 1/3`
Now
`u = 1/(x + y)``
`=> 1/(x + y) =1/ 19`
`=>x + y = 19` ....(i)
And
`v = 1/(x - y)`
`=> 1/(x -y) = 1/3`
=> x - y = 3 ...(ii)
Now adding eq i and ii
we get x = 11
And after substituting te value x in eq (ii)
we get y = 8
Hence the value oof x = 11 and y = 8
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