Solve for $$ x $$ and $$ y $$ simultaneously: $$ 2 x-y=2 $$ $$ x y=4 $$ 1.3 * Show that $$ 2.5^{n}-5^{n+1}+5^{n+2} $$ is even for all positive integer values of

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**Step 1. Solve the system of equations**

We are given:
$$
2x - y = 2
$$
$$
xy = 4
$$

1. Solve the first equation for $$ y $$:
   $$
   y = 2x - 2.
   $$

2. Substitute $$ y = 2x - 2 $$ into the second equation:
   $$
   x(2x - 2) = 4.
   $$
   
3. Expand and simplify:
   $$
   2x^2 - 2x = 4 \quad \Longrightarrow \quad 2x^2 - 2x - 4 = 0.
   $$
   
4. Divide the entire equation by $$2$$:
   $$
   x^2 - x - 2 = 0.
   $$
   
5. Factor the quadratic:
   $$
   (x - 2)(x + 1) = 0.
   $$
   
6. Solve for $$ x $$:
   - If $$ x - 2 = 0 $$, then $$ x = 2 $$.
   - If $$ x + 1 = 0 $$, then $$ x = -1 $$.

7. Find the corresponding $$ y $$-values using $$ y = 2x - 2 $$:
   - For $$ x = 2 $$:
     $$
     y = 2(2) - 2 = 4 - 2 = 2.
     $$
   - For $$ x = -1 $$:
     $$
     y = 2(-1) - 2 = -2 - 2 = -4.
     $$

Thus, the solutions are:
$$
(x,y) = (2,2) \quad \text{and} \quad (x,y) = (-1,-4).
$$

---

**Step 2. Show that $$\,2\cdot5^{n}-5^{n+1}+5^{n+2}\,$$ is even for all positive integers $$ n $$**

Given the expression:
$$
2\cdot5^{n} - 5^{n+1} + 5^{n+2},
$$
follow these steps:

1. Factor out the common factor $$ 5^{n} $$:
   $$
   2\cdot5^{n} - 5^{n+1} + 5^{n+2} = 5^{n}\left(2 - 5 + 5^2\right).
   $$
   
2. Simplify the expression inside the parentheses:
   $$
   2 - 5 + 5^2 = 2 - 5 + 25 = 22.
   $$

3. Hence, the expression becomes:
   $$
   5^{n} \cdot 22.
   $$

4. Since $$ 22 $$ is even, the product $$ 22 \cdot 5^{n} $$ is even for any integer $$ n $$.

Thus, $$ 2\cdot5^{n}-5^{n+1}+5^{n+2} $$ is even for all positive integers $$ n $$.
The solutions are $$ (x, y) = (2, 2) $$ and $$ (x, y) = (-1, -4) $$. Additionally, the expression $$ 2\cdot5^{n} - 5^{n+1} + 5^{n+2} $$ is even for all positive integers $$ n $$.

Solve for and simultaneously:

1.3 * Show that is even for all positive integer values of

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Solve for and simultaneously: 1.3 * Show that is even for all positive integer values of