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.

(A) Let the length and breadth of the rectangle be $x$ units and $y$ units respectively.
Area $= x y$
According to the first condition:
$(x - 5)(y + 3) = xy - 9$
$xy + 3x - 5y - 15 = xy - 9$
$3x - 5y = 6$ ...$(1)$
According to the second condition:
$(x + 3)(y + 2) = xy + 67$
$xy + 2x + 3y + 6 = xy + 67$
$2x + 3y = 61$ ...$(2)$
Solving equations $(1)$ and $(2)$ by the cross-multiplication method:
$\frac{x}{(-5)(-61) - (3)(-6)} = \frac{y}{(-6)(2) - (-61)(3)} = \frac{1}{(3)(3) - (-5)(2)}$
$\frac{x}{305 + 18} = \frac{y}{-12 + 183} = \frac{1}{9 + 10}$
$\frac{x}{323} = \frac{y}{171} = \frac{1}{19}$
$x = \frac{323}{19} = 17$
$y = \frac{171}{19} = 9$
Thus,the length is $17$ units and the breadth is $9$ units.