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The system of equations provided is:

\[
\begin{cases}
2x + 7y = 8 \\
y + 2x = 14
\end{cases}
\]

To solve this system, we can use either the substitution method or the elimination method. I'll proceed with the **elimination method** here:

1. Rewrite the equations for clarity:
   \[
   2x + 7y = 8 \quad \text{(Equation 1)}
   \]
   \[
   2x + y = 14 \quad \text{(Equation 2)}
   \]

2. Subtract Equation 2 from Equation 1:
   \[
   (2x + 7y) - (2x + y) = 8 - 14
   \]
   Simplifying gives:
   \[
   6y = -6
   \]

3. Solve for \( y \):
   \[
   y = -1
   \]

4. Substitute \( y = -1 \) back into Equation 2:
   \[
   2x + (-1) = 14
   \]
   Simplify:
   \[
   2x - 1 = 14
   \]
   \[
   2x = 15
   \]
   \[
   x = 7.5
   \]

Thus, the solution to the system is:
\[
x = 7.5, \quad y = -1
\]

Would you like more details or have any questions?

Here are some related questions:
1. What is the graphical interpretation of the solution \(x = 7.5\) and \(y = -1\)?
2. How would the solution change if one of the equations was modified slightly?
3. What are the possible methods to check if this solution is correct?
4. Can you explain the substitution method for solving this system?
5. How does the solution to this system compare to solving a single linear equation?

**Tip**: Always verify the solution by substituting it back into the original equations to ensure correctness!

Solve the system of equations: 2x + 7y = 8 and y + 2x = 14.

Learn how to solve the system of linear equations 2x + 7y = 8 and y + 2x = 14 using the elimination method. This example demonstrates key concepts in algebra, including solving systems of equations and eliminating variables.

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