Q:

How to teach addition of negative numbers? 

I have a friend with dyscalculia and was teaching her some some mathematics (namely, solving a linear equation, simplifying certain expressions, and what (affine linear) functions are). 
She understood solving equations of the form $ax + b = 0$ by first adding $-b$ to both sides and then diving by $a$. Dealing with negative $a$ and with expression $b - b$ was something of a problem, but I hope she figured it out, also. 
Adding slightly more complexity created more problems.  For example: $2x + 3 = -7$. We subtract 3 from both sides, getting $2x = -7 - 3$. She has great trouble seeing that $-7-3 = -10$.
How to communicate and teach the concepts here?  I tried using the thermometer analogy, explaining how $a - a = a + (-a) = 0$ and, somewhat poorly, that $-7 - 3 = - (7 + 3) = -10$. How to justify the last attempt in a useful way?  What other models or intuitions are there for understanding the negative numbers and particularly summing them? 

A:

It would help to know a little more about the nature of your friend's dyscalclia.  The sense of the integer line comes from the inferior parietal cortex (and causes difficulty with problems like "what number is halfway between 7 and 11?")  while rote memory-type problems ("what is 4 times 7?")  are associated with the basal ganglia.  It is possible to have trouble with one but not the other. 
If your friend has difficulty with rote memory, you need to invoke their visual intuition (that is, rely heavily on a number line).  Because the inferior parietal cortex is associative (that is, it's located in the brain at a convergence of senses) I would recommend also including their kinesthetic sense by making a physical number line and having them practice by walking along it to do number problems (I use playing cards, black for positive and red for negative). 
If your friend instead has difficulty with mathematical intuition, I would present a systemized set of rules that can be incorporated into rote memory.  I wouldn't stop there though, if they can master the rules that way -- I would then go back to associating those rules with mathematical intutition.  It is possible to teach someone mathematical intuition who is normally impaired this way; this is a technique using neuroplasticity which involves "rewiring" around problematic parts in the brain. 

A:

You can think of them as numbers with directions.  Draw a line with the integers on it, adding a positive number $b$ to a given one $a$ is counting $b$ positions to the right starting from $a$ (no matter if it is positive or negative), if $b$ is negative, you count towards the left. 
This point of view will also help when dealing with multiplication of negative numbers, thinking as -1 as "turning around" makes easy to understand $(-1)\times (-1) = 1 $.  In my experience the interpretation of negative numbers as "debts" never works well. 

A:

You could think of integers as pairs of natural numbers $(a,b)$ modulo the equivalence
$$(a,b) \sim (c,d)\quad \mathrm{if}\quad  a+d = b+c.$$
(In other words, you think of $(a,b)$ secretely as $a-b$.)  Addition is just entrywise addition of natural numbers:
$$(a,b)+(c,d) = (a+c,b+d).$$
Negative numbers are numbers of the form $(0,a)$.  Now the equation $-7 -3 = -10$ is
$$(0,7) + (0,3) = (0,10).$$
In summary, you can restrict yourself to adding natural numbers, provided that you consider pairs. 
I'm not sure this helps in your friend's case, though. 
Edit
Negative numbers do not arise in nature, of course.   It is the result of comparison.   The above set up with pairs is precisely that.   $(a,b)$ is positive if $a>b$, negative if $a<b$ and zero if $a=b$.  This could perhaps be woven into some sort of narrative for your friend.