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Form the pair of linear equations in the following problems and find their solutions (if they exist) by any algebraic method : (v) 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

Q4.    Form the pair of linear equations in the following problems and find their solutions (if they exist) by any algebraic method :

                (v) 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.

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Let l be the length of the rectangle and b be the width,

Now, According to the question,

(l-5)(b+3)=lb-9

\Rightarrow lb+3l-5b-15=lb-9

\Rightarrow 3l-5b-6=0..........(1)

Also,

(l+3)(b+2)=lb+67

\Rightarrow lb+2l+3b+6=lb+67

\Rightarrow 2l+3b-61=0..........(2)

By Cross multiplication method,

\frac{x}{b_1c_2-b_2c_1}=\frac{y}{c_1a_2-c_2a_1}=\frac{1}{a_1b_2-a_2b_1}

\frac{x}{(-5)(-61)-(3)(-6)}=\frac{y}{(-6)(2)-(-61)(3)}=\frac{1}{(3)(3)-(2)(-5)}

\frac{x}{305+18}=\frac{y}{-12+183}=\frac{1}{9+10}

\frac{x}{323}=\frac{y}{171}=\frac{1}{19}

x=17,\:and\:y=9

Hence the length and width of the rectangle are 17 unit and 9 unit respectively.

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