Question

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

ax + by = a²
bx + ay = b²

Answer 1

The system of the given equations may be written as

ax + by - a² = 0

bx + ay - b² = 0

Here,

`a_1 = a, b_1 = b, c_1 = -a^2`

`a_2 = b, b_2 = a, c_2 = -b^2`

By cross multiplication, we get

`=> x/(b xx (-b^2) - (-a^2) xx a) = (-y)/(a xx(-b^2) - (-a^2) xx b) = 1/(axxa - bxxb)`

`=> x/(-b^3 + a^3) = (-y)/(-ab^2 + a^2b) = 1/(a^2 - b^2)`

Now

`x/(-b^3 + a^3) = 1/(a^2 - b^2)`

`=> x = (a^3 - b^3)/(a^2 - b^2)`

`= ((a- b)(a^2 + ab + b^2))/((a- b)(a + b))`

`= (a^2 + ab + b^2)/(a + b)`

And

`(-y)/(-ab^2 + a^2b) = 1/(a^2 - b^2)`

`=> -y = (a^2b - ab^2)/(a^2 - b^2)`

`=> y = (ab^2 - a^2b)/(a^2 - b^2)`

`=> (ab(b -a))/((a-b)(a + b))`

`(-ab(a - b))/((a - b)(a + b))`

`= (-ab)/(a + b)`

Hence `x = (a^2 + ab + b^2)/(a + b), y = (-ab)/(a + b)` is the solution of the given system of the equations.