If a rectangular garden were 2 feet wider and 3 feet longer, it would be 64 square feet larger. If it were 3 feet wider and 2 feet longer, it would be 68 square feet larger. What are the dimensions of the garden?

Answer 1

Let the original dimensions of the garden be x feet by y feet.
According to the first condition, if the garden were 2 feet wider and 3 feet longer, its new dimensions would be (x+2) feet by (y+3) feet.
Given that the new area is 64 square feet larger than the original area, we can set up the equation: (x+2)(y+3) = xy + 64.
Expanding the left side gives xy + 2y + 3x + 6 = xy + 64.
Simplifying the equation, we get 3x + 2y = 58. ----(Equation 1)

According to the second condition, if the garden were 3 feet wider and 2 feet longer, its new dimensions would be (x+3) feet by (y+2) feet.
Given that the new area is 68 square feet larger than the original area, we can set up the equation: (x+3)(y+2) = xy + 68.
Expanding the left side gives xy + 3y + 2x + 6 = xy + 68.
Simplifying the equation, we get 2x + 3y = 62. ----(Equation 2)

To solve the system of equations (Equation 1 & Equation 2), we can multiply Equation 1 by 3 and Equation 2 by 2:
9x + 6y = 3 * 58,
4x + 6y = 2 * 62.

Subtracting the second equation from the first eliminates the y-term:
(9x + 6y) - (4x + 6y) = 174 - 124,
5x = 50,
x = 10.

Plugging the value of x into Equation 1 gives:
3(10) + 2y = 58,
30 + 2y = 58,
2y = 58 - 30,
2y = 28,
y = 14.

Therefore, the dimensions of the original garden are 10 feet by 14 feet.