Question

Solve for x and y:
`x/a + y/b = 2, ax – by = (a^2 – b^2)`

Answer 1

The given equations are:
`x/a + y/b = 2`
`⇒(bx+ay)/(ab)` = 2 [Taking LCM]
⇒bx + ay = 2ab                                           …….(i)
Again, ax – by =` (a^2 – b^2)`                        …..(ii)
On multiplying (i) by b and (ii) by a, we get:
`b^2x + bay = 2ab^2`                                     ……..(iii)
`a^2x – bay = a(a^2 – b^2)`                            …….(iv)
On adding (iii) from (iv), we get:
`(b^2 + a^2)x = 2a^2b + a(a^2 – b^2)`
`⇒(b^2 + a^2)x = 2ab^2 + a^3 – ab^2`
`⇒(b^2 + a^2)x = ab^2 + a^3`
`⇒(b^2 + a^2)x = a(b^2 + a^2)`
`⇒x =( a(b^2+ a^2))/((b^2+ a^2)) = a`

On substituting x = a in (i), we get:
ba + ay = 2ab
⇒ ay = ab
⇒y = b
Hence, the solution is x = a and y = b.