Question

Solve the system of equations by using the method of cross multiplication:

`(ax)/b - (by)/a - (a + b) = 0, ax - by - 2ab = 0`

Solve the system of equations by using the method of cross multiplication:

`(ax)/b - (by)/a = a + b, ax - by = 2ab`

Answer 1

The given equations may be written as:

`(ax)/b - (by)/a - (a + b) = 0`   ...(i)

ax – by – 2ab = 0   ...(ii)

Here, `a_1 = a/b, b_1 = (-b)/a, c_1 = -(a + b), a_2 = a, b_2 = -b` and `c_2 = -2ab`

By cross multiplication, we have:

∴ `x/[(-b/a) xx (-2ab) - (-b) xx (-(a + b)]) = y/([-(a + b) xx a -(-2ab) xx a/b]) = 1/([a/b xx (-b) - a xx (-b/a)])`

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

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

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

⇒ `x/(-b(a - b)) = y/(a(a - b)) = 1/(-(a - b))`

⇒ `x = (-b(a - b))/(-(a - b)) = b, y = (a(a - b))/(-(a - b)) = -a`

Hence, x = b and y = –a is the required solution.