Let's simplify the expression step by step:

$13 - 10 + 6 - 12 + 3$

1. Start with $13 - 10 = 3$
2. Then, $3 + 6 = 9$
3. Next, $9 - 12 = -3$
4. Finally, $-3 + 3 = 0$

So, the result is:

$13 - 10 + 6 - 12 + 3 = 0$

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

Here are some related questions you might find interesting:

1. How do you solve an expression with multiple additions and subtractions?
2. What is the order of operations in arithmetic expressions?
3. How do you handle negative numbers in calculations?
4. What happens when you mix addition and subtraction in one expression?
5. What are some tips for simplifying long arithmetic expressions?

Tip: When simplifying long expressions, group operations together step by step to avoid mistakes!