Solve the system of linear equations by elimination.
\[
\begin{array}{l}
x+2 y=13 \\
-x+y=5
\end{array}
\]

Step 1 ：Understand the problem: We are given a system of two linear equations and we are asked to solve it using the method of elimination.

Step 2 ：Set up the equations: The system of equations is given as: \(x + 2y = 13\) and \(-x + y = 5\).

Step 3 ：Solve the system: To use the method of elimination, we add the two equations together. This gives: \(x + 2y - x + y = 13 + 5\), which simplifies to: \(3y = 18\).

Step 4 ：Solve for y: To solve for y, we divide both sides of the equation by 3: \(y = 18 / 3\), so \(y = 6\).

Step 5 ：Substitute y into one of the original equations: Now that we have a value for y, we can substitute it into one of the original equations to solve for x. This gives: \(x + 2(6) = 13\), which simplifies to: \(x + 12 = 13\), so \(x = 1\).

Step 6 ：Check the solution: To ensure our solution is correct, we can substitute \(x = 1\) and \(y = 6\) into the second equation: \(-x + y = 5\), which simplifies to: \(-1 + 6 = 5\), so the left-hand side equals the right-hand side, confirming our solution is correct.

Step 7 ：Write the final answer: The solution to the system of equations is \(\boxed{x = 1}\) and \(\boxed{y = 6}\).