The numerator of a fraction is $4$ less than the denominator. If the numerator is decreased by $2$ and the denominator is increased by $1,$ then the denominator becomes eight times the numerator. Find the fraction.

The sum of a two-digit number and the number obtained by interchanging its digits is $110$. If the number obtained by subtracting $10$ from the original number exceeds five times the sum of the digits of the original number by $4$,then find the original number.

Find the value of $k$ under the given condition for the following pair of linear equations: $2x - 3y = 1$ and $kx + 5y = 7$,such that the system has a unique solution.

In a two-digit number, the digit at the unit's place is $(2x - 1)$ and the digit at the ten's place is $(2x + 1)$. Then, that number is:

$x=-1$ and $y=-2$ is one of the solutions of the equation $\ldots \ldots \ldots \ldots$

Form the pair of linear equations in the following problems and find their solutions (if they exist) by any algebraic method.
The area of a rectangle gets reduced by $9$ square units,if its length is reduced by $5$ units and breadth is increased by $3$ units. If we increase the length by $3$ units and the breadth by $2$ units,the area increases by $67$ square units. Find the dimensions of the rectangle.