If you are providing the car with 10N and the force of friction requires you to exceed 10N to move the car, then if you're providing the same amount of force of friction, you will not move.
In order to move the car you NEED to provide more force than the opposing force in your situation (which is friction).
Once you overcome that force, let's say you apply 20N of force to get the car to move, you will have some velocity (without doing the actual calculation, let's say your velocity is 100 km/h).
Now let's say you step on the pedal just enough to keep the car pushing with a force that matches the force of friction, then the car will stay at your speed of 100 km/h.
If you don't put ENOUGH acceleration, meaning the car is pushing less than 10N in this case, the car will start to slow down. If you don't supply any gas at all, the force of friction will just take over until your car reaches a halt.
If you put more gas, causing a net force of over 10N, your car will speed up... meaning it will be going faster than 100 km/h.
As you can see, if you balance the forces, your car will stay in constant speed. You could be in motion or at rest. If the car is going 0 km/h when you're applying the force equal to the friction, then you will remain at 0. If you're going 100 km/h while you're applying the force equal to friction, then you will remain at 100 km/h until you let friction slow you down, or if you provide more acceleration to the car.