It's not so much the run of the drain pipe to the main stack as it is the proximity to a vent. Every drain needs a nearby vent to allow air to get in behind the water. Think of it like a bottle of water; upend it, and you get the familiar "glug-glug-glug" of some water going out, then air coming in to break the vacuum. Stick a straw up through the mouth of the bottle into the air gap, and not only does the glugging stop, but the bottle drains very quickly. Your household plumbing works similarly.
The ICC organization's plumbing codes state that there should be a vent pipe within five feet of any drain. With your new location 35 feet away from the main stack (which is of course vented), you're too far away from that one, so you'll need to tie into another vent stack. Some codes allow for a wash basin and clothes washer to be each other's vent if they're within five feet, but this introduces new challenges. For instance, you can't have a P-trap on the sink itself as that closes off the pipe and prevents it venting; the trap must instead be downstream of the washer, and as this is a basement installation, if the main stack backs up from upper floors it will do so into the wash basin, so a backflow preventer at the stack end of your new drain run is highly recommended if not required.