#### Answer

The person was driving $100$ miles per hour.

#### Work Step by Step

For this problem, we know that the fine, denoted $F$, is $400$. We can now substitute $400$ for $F$ into the equation given to solve for $x$, the speed of the car in miles per hour.
$$400 = 10(x - 65) + 50$$
First, we use distributive property to simplify the equation:
$$4000 = 10x + 10(-65) + 50$$
Multiply:
$$400 = 10x - 650 + 50$$
Group like terms:
$$400 = 10x + (-650 + 50)$$
Do the addition:
$$400 = 10x + (-600)$$
To add a negative number means to subtract that number:
$$400 = 10x - 600$$
Add $600$ to both sides of the equation to isolate constants to one side of the equation:
$$1000 = 10x$$
Divide both sides by $10$ to solve for $x$:
$$x = 100$$
The person was driving $100$ miles per hour.