Question

Form the pair of linear equations in the following problem and find their solution (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

Step 1: Form the linear equations

Let us assume that

The length of the rectangle be $x$ units.

The breadth of the rectangle be $y$ units.

Now, as per the question conditions

$(x–5)(y+3)=xy-9$

$\Rightarrow$$xy+3x-5y-15=xy-9$

$\Rightarrow$ $3x–5y=6\dots \dots \dots \dots \dots \left(1\right)$

$(x+3)(y+2)=xy+67$

$\Rightarrow$ $xy+2x+3y=xy+67$

$\Rightarrow$ $2x+3y=61\dots \dots \dots \dots \dots \dots \left(2\right)$

Step 2: Find the solution

By multiplying equation $\left(1\right)$ with $2$ and equation $\left(2\right)$ with $3$, and subtracting them we get,

$6x-10y-6x-9y=12-183$

$\Rightarrow$ $-19y=-171$

$\Rightarrow$ $y=9$

Substituting this equation $\left(1\right)$ we get,

$3x-\left(5\times 9\right)=6$

$\Rightarrow$ $3x=6+45$

$\Rightarrow$ $x=\frac{51}{3}$

$\Rightarrow$ $x=17$

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