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Question
Solve each of the following systems of equations by the method of cross-multiplication
`x/a + y/b = a + b`
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Solution
The given system of the equation may be written as
`1/a x xx + 1/b xx y -(a + b) = 0`
`1/a^2 x xx 1/b^2 xx y - 2 = 0`
Here
`a_1 = 1/a, b_1 = 1/b, c_1 = -(a+ b)`
`a_2 = 1/a^2, b_2 = 1/b^2 , c_2 = -2`
By cross multiplication, we ge
`=> x/(1/b xx (-2) - 1/(b^2) x - (a +b)) = (-y)/(1/\a xx -2 - 1/a^2 x - (a + b)) = 1/(1/a xx 1/b^2 - 1/a^2 xx 1/b)`
`=> x/(-2/b + a/b^2 + 1/b) = (-y)/(-2/a + 1/a + b/a^2) = 1/(-1/(ab^2) - 1/(a^2b))`
`=> x/(a/b^2 - 1/b) = (-y)/(-1/a + b/a^2) = 1/(1/(ab^2) - 1/(a^2b))`
`=> x/((a- b)/b^2) = y/((a-b)/a^2) = 1/((a - b)/(a^2b^2))`
`=> x = (a - b)/b^2 xx 1/((a - b)/(a^2b^2)) = a^2` and
`y = (a - b)/a^2 xx 1/((a - b)/(a^2b^2)) = b^2`
Hence `x = a^2, y= b^2` is solution of the given system of the equtaions.
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