Let's call the amount of money you started with in your savings account "x".
In 19 weeks, you added $5 each week, so you added a total of $5 x 19 = $95.
Adding this amount to the initial amount x, we get:
x + $95 = $115
Subtracting $95 from both sides, we get:
x = $20
Therefore, you started with $20 in your savings account.