a man wants to donate rectangular …

Ans. Let the length of Plot = x m

Breadth of plot = y m

Area of Plot = xy

According to First condition, 

(x-50) (y+50) = xy 

=> xy + 50x -50y - 2500 = xy 

=> 50x - 50y = 2500

=> x- y = 50 ........ (1)

According to 2nd condition, 

(x-10) (y-20) = xy - 5300

=> xy - 20x - 10y + 200 = xy -  5300

=> -20x - 10y = -5500

=> 20x + 10y = 5500

=> 2x + y = 550 ........(2) 

We can represent these two equation using Matrix 

A = {tex}\begin{bmatrix} 1 & -1 \\ 2& 1 \end{bmatrix}{/tex},  |A| = 1+2 = 3

X = {tex}\begin{bmatrix} x \\ y \end{bmatrix}{/tex}

B = {tex}\begin{bmatrix} 50 \\ 550 \end{bmatrix}{/tex}

Such that AX = B

X = A^-1B
A^-1 = {tex}Adjoint \space of \space A\over Determinant \space of \space A {/tex}

=> X= {tex}{1\over 3} \begin{bmatrix} 1 & 1 \\ -2& 1 \end{bmatrix}\begin{bmatrix} 50 \\ 550 \end{bmatrix}{/tex}

=> X = {tex}{1\over 3}\begin{bmatrix} 600 \\ 450 \end{bmatrix}{/tex}

=>X =  {tex}\begin{bmatrix} 200 \\ 150 \end{bmatrix}{/tex}

x = 200

y = 150 

So Length of Plot = 200 m

Breadth of Plot = 150 m

He wants to donate land for school because he know the worth of being educated.