Matthew K. answered • 05/28/21

The first step is to extract the equations from the problem. Here we have (l is length of garden and w is width):

(1) l = 2 * w - 3

(2) (l + 3) * (w + 3) = 108

**I am ignoring units because it will be less readable. Everything is in units of ft or ft^{2}, whichever fits the equation.

The second equation is a little more difficult to extract from the problem. We know the area of a rectangle of width of width of the garden plus 3ft (w' = w +3) and length of length of the garden plus 3ft (l' = l + 3) where w' and l' are the width and length of the garden and path together. Then these values are plugged into the area of a rectangle to get (2).

Here, we already know the relationship between l and w from (1) so we can plug (1) into (2) and simplify using the quadratic formula to find w.

(3) ([2 * w - 3] + 3) * (w + 3) = 108

(4) (2 * w) * (w + 3) = 108

(5) w^{2} + 3 * w = 54

(6) w^{2} + 3 * w - 54 = 0

(7) w = (-3 +/- sqrt(9 - 4 * 1 * -54)) / 2

(8) w = (-3 + 15) / 2 = 6 OR w = (-3 - 15) / 2 = -9

From these two solutions, only w = 6 makes sense since we cannot have a negative width to the garden. Now we plug in w = 6 to (1) to find l.

(9) l = 2 * 6 - 3 = 9

Now we have the dimensions of the garden. w = 6 ft and l = 9 ft.