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
