Yes a solution of vinegar Epsom salt and Dawn dish soap does kill weeds.
The weeds won't die immediately but within a day or two of spraying the weeds with the vinegar Epsom salt and Dawn dish soap the weeds should begin wilting and die off.
It can take a few days to a week for the weeds to fully die but the weeds will eventually die.
You can also use regular salt instead of Epsom Salt in the mixture of vinegar and dawn dish soap.
Even straight vinegar kills weeds but the vinegar works better with some soap and salt.
You can also use some diesel fuel but if you do it can kill grass and other plants.
The vinegar and salt and dawn dish soap is much better for the environment though.