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Linear Equations in Two Variables — Maths STD 10 — Question

Gujarat BoardEnglish MediumSTD 10MathsLinear Equations in Two Variables4 Marks
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

Let the length and breadth of the rectangle be $x$ and $y$ units respectively.
Then area of rectangle = xy square units
It is given that if length is increared and breadth reduced each by $2$ units. then the area is reduced by $28$ square units.
$\Rightarrow (x + 2) (y - 2) = xy - 28$
$\Rightarrow xy - 2x + 2y - 4 = xy - 28$
$\Rightarrow -2x + 2y - 4 + 28 = 0$
$\Rightarrow -2x + 2y + 24 = 0$
$\Rightarrow 2x - 2y - 24 = 0 ......(i)$
It is also given that the length is reduced by $1$ unit and breath is increcred by $2$ units then the area is increared by $33$ square units.
$\Rightarrow (x - 1) (y + 2) = xy + 33$
$\Rightarrow xy + 2x - y - 2 = xy + 33$
$\Rightarrow xy + 2x - y - 2 - xy - 33 = 0$
$\Rightarrow 2x - y - 35 = 0 ......(ii)$
Now, subtracting eq. $(ii)$ from eq. $(i)$ and we get
$\Rightarrow 2x - 2y - 24 - (2x - y - 35) = 0$
$\Rightarrow 2x - 2y - 24 - 2x + y + 35 = 0$
$\Rightarrow -y + 11 = 0$
$\Rightarrow y = 11$
Putting the value of $y$ in eq.$ (i)$
$\Rightarrow 2x - 2y - 24 = 0$
$\Rightarrow 2x - 2 \times 11 - 24 = 0$
$\Rightarrow 2x - 22 - 24 = 0$
$\Rightarrow 2x - 46 = 0$
$\Rightarrow x = 23$
Hence, the length of the rectangle is $23$ and breadth of the rectangle is $11$
Area os rectangle = length \times breadth
$= 23 \times 11$
$= 253$ square units.

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