Solve the following system of equations over the real numbers:
$2x_1 = x_5 ^2 - 23$
$4x_2 = x_1 ^2 + 7$
$6x_3 = x_2 ^2 + 14$
$8x_4 = x_3 ^2 + 23$
$10x_5 = x_4 ^2 + 34$
1. **Given system of equations:**
   \[
   \begin{cases}
   2x_1 = x_5^2 - 23 \\
   4x_2 = x_1^2 + 7 \\
   6x_3 = x_2^2 + 14 \\
   8x_4 = x_3^2 + 23 \\
   10x_5 = x_4^2 + 34
   \end{cases}
   \]

2. **Assume the last expression is correct:**
   \[
   10x_5 = x_4^2 + 34
   \]

3. **Add all the equations:**
   \[
   2x_1 + 4x_2 + 6x_3 + 8x_4 + 10x_5 = (x_5^2 - 23) + (x_1^2 + 7) + (x_2^2 + 14) + (x_3^2 + 23) + (x_4^2 + 34)
   \]

4. **Simplify the right-hand side:**
   \[
   2x_1 + 4x_2 + 6x_3 + 8x_4 + 10x_5 = x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 + 55
   \]

5. **Rearrange the equation:**
   \[
   x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 - 2x_1 - 4x_2 - 6x_3 - 8x_4 - 10x_5 + 55 = 0
   \]

6. **Complete the square for each term:**
   \[
   (x_1 - 1)^2 + (x_2 - 2)^2 + (x_3 - 3)^2 + (x_4 - 4)^2 + (x_5 - 5)^2 = 0
   \]

7. **Since the sum of squares is zero, each square must be zero:**
   \[
   \begin{cases}
   x_1 - 1 = 0 \\
   x_2 - 2 = 0 \\
   x_3 - 3 = 0 \\
   x_4 - 4 = 0 \\
   x_5 - 5 = 0
   \end{cases}
   \]

8. **Solve for each variable:**
   \[
   \begin{cases}
   x_1 = 1 \\
   x_2 = 2 \\
   x_3 = 3 \\
   x_4 = 4 \\
   x_5 = 5
   \end{cases}
   \]

The final answer is \( \boxed{ (x_1, x_2, x_3, x_4, x_5) = (1, 2, 3, 4, 5) } \).