To solve the simultaneous equations:

$x - 2y = 13$ $2x + y = 11$

Step 1: Solve one equation for one variable

From the second equation, solve for $y$: $y = 11 - 2x$

Step 2: Substitute $y = 11 - 2x$ into the first equation

Substitute $y = 11 - 2x$ into $x - 2y = 13$: $x - 2(11 - 2x) = 13$

Simplify: $x - 22 + 4x = 13$ $5x - 22 = 13$

Solve for $x$: $5x = 35$ $x = 7$

Step 3: Substitute $x = 7$ into $y = 11 - 2x$

$y = 11 - 2(7)$ $y = 11 - 14$ $y = -3$

Final Solution:

$x = 7, \, y = -3$

Let me know if you'd like more details!

Here are five related questions to deepen your understanding:

1. How would the solution change if one equation were modified?
2. Can you solve the same problem graphically and interpret the solution?
3. What happens if there is no solution to a system of equations?
4. How can simultaneous equations be solved using matrices?
5. What if the equations represented parallel lines? What would that mean?

Tip: Always double-check your substitution step to ensure accuracy in simultaneous equations!