Hi so my brother has a 1979 Mazda Rx7 bone stock with no modifications done to it. The only items replaced was the fuel pump and starter both in working order. The problem we are having is when you switch the key onto the START position it turns on for a brief second but as soon as the key goes back to the ON position the car dies.Any ideas on what can be causing this?

 

Carb gunked up, fuel delivery issues, could be a lot of things.

 

Clean the external resistors terminals. Make sure all the fuses are good and have good contact with the fuse block inside the car.

I also chased back the ignition wires coming out of the switch, I believe it is x-08 in the FSM, to make sure you don’t have a broken wire.

 

I'll try this and see if it works as for the fuel delivery I dont believe it's that because the carb bowls are getting fuel to them

 

Good possibility ignition switch contacts are worn out-or electrical section of switch is loose.
Put eyes on switch/contacts/wiring.

Put test light on positive primary terminals of both ignition coils with key in run position. There should be @9-10 volts in key run position. If this voltage is present and car is not running,investigate condition of points/condensers to be certain ignition system is functional.

Upgrading to later FB distributor is popular modification for more reliable/powerful ignition system.