Solve the following systems of simultaneous linear equations by the elimination method

$\dfrac{x}{a} - \dfrac{y}{b} = 0$

ax + by = a² + b²

$\phantom{\Rightarrow} \dfrac{x}{a} - \dfrac{y}{b} = 0 \\[1em] \Rightarrow \dfrac{bx - ay}{ab} = 0 \\[1em] \Rightarrow bx - ay = 0.$

Given equations can be written as,

bx - ay = 0 …….(i)

ax + by = (a² + b²) …….(ii)

Multiplying eq. (i) by a and (ii) by b we get,

abx - a²y = 0 ………(iii)

abx + b²y = b(a² + b²) …….(iv)

Subtracting eq. (iii) from (iv) we get,

⇒ abx + b²y - (abx - a²y) = b(a² + b²)

⇒ abx - abx + b²y + a²y = b(a² + b²)

⇒ y(b² + a²) = b(a² + b²)

⇒ y = b.

Substituting value of y in eq. (i) we get,

⇒ bx - ay = 0

⇒ bx - ab = 0

⇒ bx = ab

⇒ x = a.

Hence, x = a and y = b.