Question

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

x + ay = b
ax − by = c

Answer 1

The given system of equations may be written as

x + ay - b = 0

ax − by - c = 0

Here,

`a_1 = 1, b_1 = a, c_1 = -b`

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

By cross multiplication, we get

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

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

Now

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

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

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

And

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

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

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

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