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Question
Solve each of the following systems of equations by the method of cross-multiplication :
`ax + by = (a + b)/2`
3x + 5y = 4
Solution
The given system of equation is
`ax + by = (a + b)/2` .....(i)
3x + 5y = 4 ....(ii)
From (i), we get
2(ax + by) = a + b
`=> 2ax + 2by - (a + b) = 0` .....(iii)
From (ii), we get
3x + 5y - 4 = 09
here
`a_1 = 2a, b_1 = 2b, c_1 = -(a + b)`
`a_2 = 3, b_2 = 5, c_2 = -4`
By cross multiplication, we hav-04e
`=> x/(2b xx (-4)- [-(a + b)]xx5) = (-y)/(2a xx (-4) - [-(a + b)] xx 3) = 1/(2a xx 5 - 2b xx 3) `
`=> x/(-8b + 5(a + b)) = (-y)/(-8a + 3(a + b)) = 1/(10a - 6b)`
`=> x/(-8b + 5a + 5b) = (-y)/(-8a + 3a + 3b) = 1/(10a - 6b)`
`=> x/(5a - 3b) = (-y)/(-5a + 3b) = 1/(10a - 6b)`
Now
`x/(5a - 3b) = (-y)/(-5a + 3b) = 1/(10a - 6b)`
`=> x = (5a -3b)/(10a - 6b) = (5a - 3b)/(2(5a - 3b)) = 1/2`
And
`(-y)/(-5a + 3b) = 1/(10a - 6b)`
`=> -y = (-5a + 3b)/(2(5a - 3b))`
`=> y = (-(-5a + 3b))/(2(5a - 3b))`
`= (5a - 3b)/(2(5a - 3b))`
`=> y = 1/2`
Hence x = 1/2, y = 1/2 is the solution of the given system of equations
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