Let's use algebra to solve this problem. Let x be the amount of money Alice had at first.
If she wanted to buy 10 markers but was short of $5.20, it means the cost of 10 markers was x - 5.20.
Since she decided to buy only 6 markers, she spent 6/10 of the total cost, which is (6/10)*(x - 5.20) = 0.6x - 3.12.
She used the remaining $4.40 to buy lunch, so we can set up an equation:
0.6x - 3.12 + 4.40 = x
Simplifying this equation, we get:
0.4x = 7.52
x = 18.80
Therefore, Alice had $18.80 at first.