Solve the following system of equations to find the coordinates (x, y) and determine the plot's dimensions: 1) 5^(x+2) + 7^(y+1) = 3468, 2) 7^y = 5^x - 76. The plot has L = W + 2 and Area = 35 sq mete

$$5^x \cdot 5^2 + 7^y \cdot 7^1 = 3468$$ $$25 \cdot 5^x + 7 \cdot 7^y = 3468$$

Let $A = 5^x$ and $B = 7^y$. The system of equations becomes:

$$25A + 7B = 3468 \quad (3)$$ $$B = A - 76 \quad (4)$$

$$25A + 7(A - 76) = 3468$$ $$25A + 7A - 532 = 3468$$ $$32A = 3468 + 532$$ $$32A = 4000$$ $$A = \frac{4000}{32}$$ $$A = 125$$

$$B = 125 - 76$$ $$B = 49$$

Since $A = 5^x$:

$$5^x = 125$$ $$5^x = 5^3$$ $$x = 3$$

Since $B = 7^y$:

$$7^y = 49$$ $$7^y = 7^2$$ $$y = 2$$

The hidden treasure's location is $\boxed{\mathbf{(3, 2)}}$.

Let the width of the plot be $W$ meters and the length be $L$ meters.
Given that the length is 2 meters longer than its width:

$$L = W + 2$$

Given that the total area of the plot is 35 square meters:

$$L \times W = 35$$

Substitute the expression for $L$ into the area equation:

$$(W + 2)W = 35$$ $$W^2 + 2W = 35$$ $$W^2 + 2W - 35 = 0$$

Factor the quadratic equation:

$$(W + 7)(W - 5) = 0$$

This gives two possible values for $W$: $W = -7$ or $W = 5$.
Since width cannot be negative, $W = 5$ meters.

$$L = W + 2 = 5 + 2 = 7 \text{ meters}$$

The dimensions of the plot are 7 meters by 5 meters.

The perimeter $P$ of a rectangle is $2(L + W)$:

$$P = 2(7 + 5)$$ $$P = 2(12)$$ $$P = 24 \text{ meters}$$

Henry plans to use a spacing of 1m from each tree around the perimeter.

$$\text{Number of trees} = \frac{\text{Perimeter}}{\text{Spacing}}$$ $$\text{Number of trees} = \frac{24 \text{ m}}{1 \text{ m/tree}}$$ $$\text{Number of trees} = 24 \text{ trees}$$

Each tree costs Ugx 2,500.

$$\text{Total Cost} = \text{Number of trees} \times \text{Cost per tree}$$ $$\text{Total Cost} = 24 \times 2500$$ $$\text{Total Cost} = 60000$$

The total amount of money needed to be spent on buying trees is $\boxed{\mathbf{\text{Ugx } 60,000}}$.