Question

If in a rectangle, the length is increased and breadth reduced each by 2 units, the area is reduced by 28 square units. If, however the length is reduced by 1 unit and the breadth increased by 2 units, the area increases by 33 square units. Find the area of the rectangle.

Answer 1

Let the length and breadth of the rectangle be and units respectively

Then, area of rectangle square units

If length is increased and breadth reduced each by units, then the area is reduced by square units

$$\begin{gathered} \left(x+2\right)\left(y-2\right)=xy-28 \\ \Rightarrow xy-2x+2y-4=xy-28 \\ \Rightarrow -2x+2y-4+28=0 \\ \Rightarrow -2x+2y+24=0 \\ \Rightarrow 2x-2y-24=0 \end{gathered}$$

Therefore,

Then the length is reduced by unit and breadth is increased by units then the area is increased by square units

$$\begin{gathered} \left(x-1\right)\left(y+2\right)=xy+33 \\ \Rightarrow xy+2x-y-2=xy+33 \\ \Rightarrow 2x-y-2-33=0 \\ \Rightarrow 2x-y-35=0 \end{gathered}$$

Therefore, $2x-y-35=0.....\left(ii\right)$

Thus we get the following system of linear equation

$$\begin{gathered} 2x-2y-24=0 \\ 2x-y-35=0 \end{gathered}$$

By using cross multiplication, we have

and

The length of rectangle is units.

The breadth of rectangle is units.

Area of rectangle =lengthbreadth,

square units

Hence, the area of rectangle is square units