Question

Question 4 (v)
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.

Answer 1

Let length and breadth of the rectangle be x units and y units respectively.
Area = xy
According to the question,
(x - 5) (y + 3) = xy - 9
$\Rightarrow 3x-5y-6=0...\left(i\right)$
(x + 3) (y + 2) = xy + 67
$\Rightarrow 2x+3y-61=\mathrm{0.......}\left(ii\right)$

Multiplying (i) by 2, we get
$6x-10y-12=0\Rightarrow 6x-10y=\mathrm{12....}\left(iii\right)$

Multiplying (ii) by 3, we get
$6x+9y-183=0\Rightarrow 6x+9y=\mathrm{183....}\left(iv\right)$

Subtracting (iv) from (iii), we get
$19y=171$
$\Rightarrow y=\frac{171}{19}\Rightarrow y=9$

Substituting y = 9 in (ii), we get

$2x+3\times 9-61=0\Rightarrow 2x=34\Rightarrow x=17$

Thus, the length of the rectangle is 17 units and the breadth is 9 units.