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Question
Solve the following systems of equations:
`1/(2(x + 2y)) + 5/(3(3x - 2y)) = (-3)/2`
`5/(4(x + 2y)) - 3'/(5(3x - 2y)) = 61/60`
Solution
Let `1/(x + 2y) = u and 1/(3x - 2y) = v`
Then, the given system of equation becomes
`u/2 + (5v)/3 = (-3)/2`
`=> (3u + 10v)/6 = (-3)/2`
`=> 3u + 10v = (-3 xx 6)/2`
=> 3u + 10v = -9 ....(i)
And `(5u)/4 - (3v)/5 = 61/60`
`=> (25u - 12v)/20 = 61/60`
`=> 25u- 12v = 61/3` ...(ii)
Multiplying equation (i) by 12, and equation (ii) by 10, we get
36u + 120vg = -108 ...(iii)
`250u - 120v = 610/3` .....(iv)
Adding equation (iii) and equation (iv), we get
`36u + 250uy = 610/3 - 108`
`=> 286u = (610 - 324)/3`
`=> 286u = 286/3`
=> u = 1/3
Putting u = 1/3 in equation (i) we get
`3 xx 1/3 + 10v = -9`
`=> 1 + 10v = -9`
=> 10v = -9 - 1
`=> v = (-10)/10 = -1`
Now `u = 1/(x + 2y)`
`=> 1/(x + y) = 1/3`
=> x + 2y = 3 ....(v)
And `v = 1/(3x - 2y)`
=> 1/(3x - 2y) = -1
=> 3x - 2y = -1 ......(vi)
Putting x = 1/2 in equation (v) we get
`1/2 + 2y = 3`
`=> 2y = 3 - 1/2`
`=> 2y = (6 -1)/2`
`=> y = 5/4`
Hence, solution of the given system of equations is x = 1/2, y = 5/4
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