We have two relations between length and width. One is given in the problem statement. The other is given by the formula for perimeter. We can solve the two equations in two unknowns using substitution.
Let w and l represent the width and length of the sign in feet, respectively.
... l = 2w -12 . . . . . the length is 12 ft less than twice the width
... p = 2(l +w) = 114 . . . . the perimeter is 114 ft
Using the first equation for l, we can substitute for l in the second equation.
... 114 = 2((2w -12) +w)
... 114 = 6w -24 . . . . . . . . simpify
... 138 = 6w . . . . . . . . . . . add 24
... 23 = w . . . . . . . . . . . . . divide by 6
... l = 2w -12 = 2·23 -12 = 34 . . . . use the equation for l to find l
The length and width of the sign are 34 ft and 23 ft, respectively.
