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?
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.
To solve this problem, let's assume that the original dimensions of the garden are represented by length "l" and width "w".
According to the information given, if the garden were 2 feet wider and 3 feet longer, it would be 64 square feet larger. This can be expressed as an equation:
(l + 3)(w + 2) = lw + 64 [Equation 1]
Similarly, if the garden were 3 feet wider and 2 feet longer, it would be 68 square feet larger. This can be expressed as another equation:
(l + 2)(w + 3) = lw + 68 [Equation 2]
Now, we can solve these two equations simultaneously to find the dimensions of the garden.
Step 1: Expand the equations:
(lw + 3w + 2l + 6) = lw + 64 [Expanding Equation 1]
(lw + 2w + 3l + 6) = lw + 68 [Expanding Equation 2]
Step 2: Simplify the equations:
3w + 2l + 6 = 64 [Simplifying Equation 1]
2w + 3l + 6 = 68 [Simplifying Equation 2]
Step 3: Rearrange the equations:
3w + 2l = 58 [Equation 3 after rearranging Equation 1]
2w + 3l = 62 [Equation 4 after rearranging Equation 2]
Step 4: Multiply Equation 3 by 2 and Equation 4 by 3:
6w + 4l = 116 [Equation 5 after multiplying Equation 3 by 2]
6w + 9l = 186 [Equation 6 after multiplying Equation 4 by 3]
Step 5: Subtract Equation 5 from Equation 6 to eliminate "w":
(6w + 9l) - (6w + 4l) = 186 - 116
6w - 6w + 9l - 4l =70
5l = 70
Step 6: Solve for "l":
l = 70 / 5
l = 14
Step 7: Substitute the value of "l" into Equation 3 to solve for "w":
3w + 2(14) = 58
3w + 28 = 58
3w = 58 - 28
3w = 30
w = 30 / 3
w = 10
Therefore, the original dimensions of the garden are 14 feet in length and 10 feet in width.
To solve this problem, we can set up a system of equations based on the given information.
Let's assume the original dimensions of the rectangular garden are length (L) and width (W).
According to the first piece of information, if the garden were 2 feet wider (W + 2) and 3 feet longer (L + 3), it would be 64 square feet larger than the original area (L * W). Therefore, we can write the equation:
(L + 3)(W + 2) = L * W + 64 .................(Equation 1)
According to the second piece of information, if the garden were 3 feet wider (W + 3) and 2 feet longer (L + 2), it would be 68 square feet larger than the original area (L * W). Therefore, we can write the equation:
(L + 2)(W + 3) = L * W + 68 .................(Equation 2)
Now, we have a system of two equations with two variables (L and W). We can solve this system to find the values of L and W.
Let's simplify the equations and solve them:
Expanding Equation 1:
(L + 3)(W + 2) = L * W + 64
LW + 3W + 2L + 6 = LW + 64
3W + 2L = 58 ...............(Equation 3)
Expanding Equation 2:
(L + 2)(W + 3) = L * W + 68
LW + 2W + 3L + 6 = LW + 68
2W + 3L = 62 ...............(Equation 4)
Now, we have a system of linear equations:
3W + 2L = 58 ...............(Equation 3)
2W + 3L = 62 ...............(Equation 4)
We can solve this system of equations using various methods such as substitution or elimination. Here, I'll use the elimination method to simplify the solution process.
Multiplying Equation 3 by 3 and Equation 4 by 2:
9W + 6L = 174 ...............(Equation 5)
4W + 6L = 124 ...............(Equation 6)
Subtracting Equation 6 from Equation 5:
(9W + 6L) - (4W + 6L) = 174 - 124
5W = 50
W = 10
Plugging the value of W back into Equation 3:
3(10) + 2L = 58
30 + 2L = 58
2L = 58 - 30
2L = 28
L = 14
Therefore, the dimensions of the garden are 14 feet (length) and 10 feet (width).