### If the length of a rectangle is increased by $14$ units and its breadth is decreased by $7$ units then the area of the rectangle is increased by $35$ square units. However, if we decrease its length by $7$ units and increase breadth by $3$ units. Its area is decreased by $105$ square units. Find the length and breadth of the rectangle.

\begin{aligned} & \text{Length} = 35 \text{ units} \\ & \text{Breadth} = 27 \text{ units} \end{aligned}

Step by Step Explanation:
1. Let us assume the length and breadth of the rectangle be $x$ units and $y$ units respectively.
2. It is given that if the length of a rectangle is increased by 14 units and its breadth is decreased by 7 units then the area of the rectangle is increased by 35 square units.
Now, \begin{aligned} & \text{New length} = (x + 14) \text{ units} \\ & \text{New breadth} = (y - 7) \text{ units } \\ \therefore \space & \text{New area} = (x + 14)(y - 7) \text{ square units} \\ \therefore \space & (x + 14)(y - 7) - xy = 35 \\ \implies & xy - 7 x + 14 y - 98 - xy = 35 \\ \implies & 14 y - 7 x = 133 \\ \implies & 2 y - x = 19 && \ldots (i) \end{aligned}
3. Similarly, if we decrease its length by 7 units and increase breadth by 3 units. Its area is decreased by 105 square units.
Now, \begin{aligned} & \text{New length} = (x - 7) \text{ units} \\ & \text{New breadth} = (y + 3) \text{ units } \\ \therefore \space & \text{New area} = (x - 7)(y + 3) \text{ square units} \\ \therefore \space & xy - (x - 7)(y + 3) = 105 \\ \implies & xy - (xy + 3 x - 7 y - 21) = 105 \\ \implies & xy - xy - 3 x + 7 y + 21 = 105 \\ \implies & 7 y - 3 x = 84 && \ldots (ii) \end{aligned}
4. On multiplying $(i)$ by 3 we get:\begin{aligned} & 6 y - 3 x = 57 && \ldots (iii) \\ \end{aligned} Subtracting $(iii)$ from $(ii),$ we get:\begin{aligned} y = 27 \end{aligned}
5. Putting $y = 27$ in $(i),$ we get:\begin{aligned} & (2 \times 27) - x = 19 \\ \implies & x = 54 - 19 = 35 \end{aligned}
6. Hence, \begin{aligned} & \text{Length} = x \text{ units} = 35 \text{ units} \\ & \text{Breadth} = y \text{ units} = 27 \text{ units} \end{aligned} 