We pick either of the equations and write one variable in terms of the other.
Let us consider the equation bx−ay=0 and write it as
x=ayb
Substitute the value of x in equation ax+by=a2+b2. We get
a(ayb)+by=a2+b2⇒a2yb+by=a2+b2⇒a2y+b2yb=a2+b2⇒(a2+b2)y=b(a2+b2)⇒y=b
Therefore, y=b
Substituting this value of y in the equation x=ayb, we get
x=abb=a
Hence, the solution is x=a,y=b.