A farmer plans to use 21 m of fencing to enclose a rectangular pen having an area of 55 m². Only three sides of the pen need fencing because of an existing wall. Find the dimensions of the pen?

Let the length of the sides of pen at right angle to the wall be equal to x meters.  Let the length of the sides of pen  parallel to the wall be equal to y meters.  Then the length of the fence needed is 2x +y; this is equal to 21m.  Therefore, 2x+y = 21  Area of the pen is x times y  = xy; this is 55 square meter.  Therefore, xy = 55  2x+y = 21, therefore, y = 21-2x  Substituting value of y in the equation xy = 55,we have x(21-2x) = 55therefore, 21x-2x²  = 55  therefore, 0=55-21x+2x^{2therefore, 2x2 -21x+55 =0This can also be put as: (2x-11) x (x-5) = 0 Since the product of two values , i.e., (2x-11) and (x-5) is zero, either both of them or alt least one of them is equal to zero.if 2x - 11 = 0 then, x = 11/2if x-5 = 0 then, x = 5}Since 2x+y = 21, if x =11/2, then y = 21-2x11/2 = 10 If x = 5, then y = 21-2x5 = 11 Therefore, there are two possible answers. Pen is 5 1/2 x 10 or 5 x 11.