But i assume from your question that sink doesn t have any drain.
Tie washer drain into sink.
It can then run horizontally to tie into another vent as long as it maintains a 1 4 inch per foot slope toward the drain.
Do not tie it into the trap.
Laundry sink waste ties in below the standpipe tie in.
Locate your existing washer s drain line specifically where it goes down into the floor or wall.
Tee wye off the vent for the sink above the standpipe waste at least 42 off the floor and if sink on the left in this photo then above the laundry box.
Using your utility sink s drain line measurement measure up from the ground and make a mark on your washer s drain line.
The washing machine drain 1 5 would run horizontally along wall pick up the lavatory drain still 1 5 and wye into a 2 drain with the shower which then wyes into a 3 drain with the toilet and on to a sump.
Washer drain hose to utility sink.
If that s the problem then your objective will work you can tie the sink drain to the washer drain.
Water is drained from the washing machine directly into the sink.
Do not connect it directly to a drain stand pipe unless the pipe is at least 2 inches in diameter.
Connect the washing machine to the sink drain if.
The drain hose from the washing machine is hung on the edge of the laundry sink.
This is where you will place the new y fitting that will separate the washer s drain from the utility sink s drain in the next.
If possible and applicable note.
There are three rules.
This install is in a basement so it would be very difficult to cut more concrete or run a separate line to the sump.
This is in virginia.
Something like this or this but in reverse.
If the utility sink was properly installed it has its own drain.
Other options for washing machine drainage in garage.