Solve the following pairs of equations by reducing them to a pair of linear equations:
$\frac{5}{x-1} + \frac{1}{y-2} = 2$
$\frac{6}{x-1} - \frac{3}{y-2} = 1$

Form the pair of linear equations in the following problems and find their solutions (if they exist) by any algebraic method.
$A$ part of monthly hostel charges is fixed and the remaining depends on the number of days one has taken food in the mess. When a student $A$ takes food for $20$ days she has to pay ₹ $1000$ as hostel charges,whereas a student $B$,who takes food for $26$ days,pays ₹ $1180$ as hostel charges. Find the fixed charges and the cost of food per day.

Solve the pair of equations:
$\frac{2}{x} + \frac{3}{y} = 13$
$\frac{5}{x} - \frac{4}{y} = -2$

The ages of two friends Ani and Biju differ by $3\, years.$ Ani's father Dharam is twice as old as Ani and Biju is twice as old as his sister Cathy. The ages of Cathy and Dharam differ by $30\, years.$ Find the ages of Ani and Biju.

Solve the following pair of equations by reducing them to a pair of linear equations:
$\frac{1}{2x} + \frac{1}{3y} = 2$
$\frac{1}{3x} + \frac{1}{2y} = \frac{13}{6}$

Form the pair of linear equations for the following problem and find their solution by the substitution method.
The larger of two supplementary angles exceeds the smaller by $18^{\circ}$. Find the angles.