Hi there. My name is*****'m sorry that you're having this problem. Many of the new printers today will not print if even one color is out. This is by design as continuing to print with low ink or an empty cartridge can allow air into the ink delivery system which can then dry out any ink left in the nozzles and cause printing problems later even with a new cartridge.
If you simply MUST print with empty cartridges, here is a trick that will allow you to do so. You'll need black electrical tape. Brother printers check the ink levels by shining a sensor light at a special “window” in the cartridges, similar by design in all their printers. Remove the empty ink cartridge(s), press it against the palm of your hand, holding it so window protrudes, then place black electrical tape over window. The light from the sensor is now reflected back, and the printer thinks the cartridge is full!
Warning: Remember to always specify that you want to print in “greyscale” in the printer settings menu before printing with empty cartridges loaded, or it could damage the printing heads (according to Brother).
Please give this a try and let me know how it works for you. If this solves the problem, would you mind taking a moment to rate my service? You can give your rating by selecting the stars located on the top right of the question page. I'd really appreciate it! Without a positive rating I am not credited for my answer. If this doesn't solve the problem or if you have questions about the procedure be sure to let me know so we can continue troubleshooting.