Question

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

ax + by = a − b
bx − ay = a + b

Answer 1

The given system of equations is

ax + by = a − b   .....(i)

bx − ay = a + b ....(ii)

here

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

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

By cross multiplication, we get

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

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

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

Now

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

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

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

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

=> x= 1

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

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

=> -y = 1

=> y = -1

Hence x = 1, y = -1 is the solution of the given system of the equations