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Question
Solve each of the following systems of equations by the method of cross-multiplication :
`5/(x + y) - 2/(x - y) = -1`
`15/(x + y) + 7/(x - y) = 10`
where `x != 0 and y != 0`
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Solution
Let `1/(x + y) = u` and `1/(x - y) = v` Then given system of equations becomes
5u - 2v = -1
15u + 7v = 10
here
`a_1 = 5, b_1 = -2, c_1 = 1`
`a_2 = 15, b_2 = 7,c_2 = -10`
By cross multiplication, we get
`=> u/((-2)xx (-10)-1 xx7) = u/(5 xx (-10)-1 xx 15) = 1/(5xx7 - (-2) xx 15 )`
`=> u/(20 - 7) = (-v)/(-50 - 15) = 1/(35 + 30)`
`=> u/13 = (-v)/(-65) = 1/65`
`=> u/13 = v/65 = 1/65`
Now
`u/13 = 1/65`
`=> u = 13/65 = 1/5`
And
`v/65 = 1/65`
`=> v = 65/65 = 1`
Now
`u = 1/(x + y)`
`=> 1/(x + y) = 1/5` ....(i)
And
`v = 1/(x - y)`
`=> 1/(x - y) = 1`
=> x - y = 1 .....(ii)
Adding equation (i) and (ii), we get
2x = 5 + 1
=> 2x = 6
`=> x = 6/2 = 3`
Adding x = 3 in eq 2
3 - y = 1
y = 3 -1
y = 2
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