To solve the system of equations:

1. Equation 1: $x + y = 5$
2. Equation 2: $x - y = 7$

Step 1: Add the two equations

$(x + y) + (x - y) = 5 + 7$ This simplifies to: $2x = 12$ So, $x = \frac{12}{2} = 6$.

Step 2: Substitute $x = 6$ into one of the original equations

Using Equation 1: $6 + y = 5$ Solving for $y$: $y = 5 - 6 = -1$

Final Solution:

$x = 6, \quad y = -1$

Would you like more details on this process or have any questions?

Here are 5 related questions to expand your understanding:

1. What happens if you subtract the two original equations instead of adding them?
2. How can you check if the solution $x = 6$ and $y = -1$ satisfies both original equations?
3. How would you solve these equations using substitution instead of elimination?
4. What would the solution be if the equations were $x + y = 5$ and $x - y = -7$?
5. Can this method be applied to non-linear equations as well? Why or why not?

Tip: Always verify your solution by plugging the values back into the original equations.