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If the length of a rectangle is reduced by $5$ cm and its breadth is increased by $2$ cm, then the area of the rectangle is reduced by $80$ cm$^2$. However, if we increase the length by $10$ cm and decrease the breadth by $5$ cm, its area is increased by $50$ cm$^2$. Find the length and breadth of the rectangle.
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Let the length of rectangle be $x$ cm
and the breadth be $y$ cm
Area of rectangle = $xy$ ($\frac{1}{2}$)
$(x-5) (y+2) = xy - 80 \Rightarrow 2x - 5y + 70 = 0$ (1$\frac{1}{2}$)
$(x+10) (y-5) = xy + 50 \Rightarrow -5x + 10 y - 100 = 0$ (1$\frac{1}{2}$)
Solving the two equations, we get
$x = 40$ and $y = 30$ (1$\frac{1}{2}$)
$\therefore$ Length of rectangle = $40$ cm
and Breadth of rectangle = $30$ cm
and the breadth be $y$ cm
Area of rectangle = $xy$ ($\frac{1}{2}$)
$(x-5) (y+2) = xy - 80 \Rightarrow 2x - 5y + 70 = 0$ (1$\frac{1}{2}$)
$(x+10) (y-5) = xy + 50 \Rightarrow -5x + 10 y - 100 = 0$ (1$\frac{1}{2}$)
Solving the two equations, we get
$x = 40$ and $y = 30$ (1$\frac{1}{2}$)
$\therefore$ Length of rectangle = $40$ cm
and Breadth of rectangle = $30$ cm