Question

Solve each of the following systems of equations by the method of cross-multiplication :

2ax + 3by = a + 2b
3ax + 2by = 2a + b

Answer 1

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.