Form the pair of linear equations in the following problems and find their solutions (if they exist) by any algebraic method. $A$ fraction becomes $\frac{1}{3}$ when $1$ is subtracted from the numerator and it becomes $\frac{1}{4}$ when $8$ is added to its denominator. Find the fraction.

For which values of $a$ and $b$ does the following pair of linear equations have an infinite number of solutions?
$2x + 3y = 7$
$(a-b)x + (a+b)y = 3a + b - 2$

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

On comparing the ratios $\frac{a_{1}}{a_{2}}, \frac{b_{1}}{b_{2}}$ and $\frac{c_{1}}{c_{2}},$ find out whether the lines representing the following pairs of linear equations intersect at a point,are parallel or coincident:
$6x - 3y + 10 = 0$
$2x - y + 9 = 0$

Form the pair of linear equations for the following problem and find their solution by the substitution method.
The taxi charges in a city consist of a fixed charge together with the charge for the distance covered. For a distance of $10 \, km$,the charge paid is ₹ $105$ and for a journey of $15 \, km$,the charge paid is ₹ $155$. What are the fixed charges and the charge per $km$? How much does a person have to pay for travelling a distance of $25 \, km$?

Solve the following pair of linear equations by the elimination method and the substitution method:
$x+y=5$ and $2x-3y=4$