Question

Solve the following system of equations by the method of cross-multiplication:

ax + by = a – b

bx – ay = a + b

Using cross-multiplication method, solve the following system of simultaneous linear equations:

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

a₁ – a, b₁ = b, c₁ = b – a

a₂ = b, b₂ = –a, c₂ = –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/(-a xx a - b xx b)`

⇒ `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 equations.