Let
#color(white)("XXX")L_L# be the length of Luke's garden
#color(white)("XXX")W_L# be the width of Luke's garden
#color(white)("XXX")A_L# we the area of Luke's garden
#color(white)("XXX")L_S# the the length of Seth's garden
#color(white)("XXX")W_S# be the width of Seth's garden
#color(white)("XXX")A_S# be the area of Seth's garden
Since we are told Seth's garden is square
#color(white)("XXX")L_S=W_S#
and
#color(white)("XXX")A_S=(L_S)^2#
Further we are told that
#color(white)("XXX")L_S=L_L-3#
So
#color(white)("XXX")A_S=(L_L-3)^2=L_L^2-6L_L+9#
We are also told that
#color(white)("XXX")W_L=L_L/2#
So
#color(white)("XXX")A_L=L_L xx W_L = (L_L^2)/2#
Since the areas are equal
#color(white)("XXX")L_L^2-6L_L+9=L_L^2/2#
#color(white)("XXX")rarr2L_L^2-12L_L+18=L_L^2#
#color(white)("XXX")rarr L_L^2-12L_L+18=0#
#color(white)("XXX")rarr (L_L-9)(L_L-3)=0#
and
either #L_L=9# or #L_L=3#
Since #L_S=L_L-3# and assuming Seth's garden is not #0# feet long
#L_L=3# must be extraneous
#color(white)("XXX")rArr L_L=9#
Luke's garden
Since #W_L=L_L/2#
the perimeter of Luke's garden must be
#color(white)("XXX")2xx(9+9/2)=27# feet.
Seth's garden
Since #L_S=L_L-3 rarr L_S=6#
and #L_S=W_S#
the perimeter of Seth's garden must be
#color(white)("XXX")2xx(6+6)=24# feet
