Question

Solve for x and y:
$ax+by-a+b=0,bx-ay-a-b=0.$

Answer 1

The equations may be written as
ax + by = a - b ..............(i)
bx - ay = a + b ..............(ii)
Multiplying (i) by a and (ii) by b, we get:
a²x + aby = a² - ab .................(iii)
b²x - aby = ab + b² ...................(iv)
On adding (iii) and (iv), we get:
(a² + b²)x = a² + b²
$\Rightarrow x=\frac{\left(a^2+b^2\right)}{\left(a^2+b^2\right)}=1$
Putting x = 1 in (i), we get:
(a × 1) + by = a - b
⇒ a + by = a - b
⇒ by = - b ⇒ y = - 1
∴ x = 1 and y = - 1 is the required solution.