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Question
Solve each of the following systems of equations by the method of cross-multiplication :
2ax + 3by = a + 2b
3ax + 2by = 2a + b
Solution
The given system of equations is
2ax + 3by = a + 2b ....(i)
3ax + 2by = 2a + b ....(ii)
Here
`a_1 = 2a, b_1 = 3b, c_1 = -(a + 2b)`
`a_2 = 3z, b_2 = 2b, c_2 = -(2a + b) `
By cross multiplication we have
`=> x/(-3b xx (2a + b) - [-(a + 2b)]xx2b) = (-y)/(-2a xx (2a + b)-[-(a +2b)] xx 3a) = 1/(2a xx 2b - 3b xx 3a)`
`=> x/(-3b + (2a + b) + 2b (a + 2b)) = (-y)/(-2a(2a + b) + 3a (a + 2b)) = 1/(4ab - 9ab)`
`=> x/(-6ab = 3b^2 + 2ab + 4b^2) = (-y)/(-4a^2 -2ab
+ 3a^2 + 6ab) = 1/(4ab - 9ab)`
`=> x/(-4ab + b^2) = (-y)/(-a^2 + 4ab) = 1/(-5ab)`
Now
`x/(-4ab + b^2) =- 1/(-5ab)`
`=> x = (-4ab + b^2)/(-5ab)`
`=> (-b(4a - b))/(-5ab)`
`= (4a - b)/(5a)`
And `(-y)/(-a^2 + 4ab) = 1/(-5ab)`
`=> -y = (-a^2 + 4ab)/(-5ab)`
`=> -y (a - 4b)/(5b)`
`=> y = (4b -a)/(5b)`
Hence `x = (4a -b)/(5a), y = (4b - a)/(5b)` is the solution of the given system of equation.
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