I’d be surprised if ASUS didn’t include a composite cable in the video card kit - Find out what the video card is (you can usually tell when you first turn the computer on, the Video card flashes it’s model on your screen before the BIOS screens come up) - and then go to the Asus Website & download the manual for your video card
If I had to guess, I’d say the black cable on the card goes to the yellow socket on the TV.
The Audio will still require the adapter cable though - or you can just use the computer speakers in the meantime, until you get that audio adapter cable.
The red, green and blue (black) cables are commonly for COMPONENT VIDEO (which is 3 seperate parts of a high quality video signal)
The Yellow, Red & White cables are for COMPOSITE (all parts of the video combined) and Stereo Audio.
Composite Video = Yellow, Left & Right Audio = Red & White plugs.
You will need a 3.5mm Stereo Audio to Stereo RCA adapter (that’s the tiny black plug with long metal bit and two rubber bands, to the White & Red RCA plugs you already have) - if you pay more than $5US for it (including P&H), you’re being ripped off.
The Black plug goes into the little green socket on your sound card.
Does your TV have a S-Video Input? If not, you can get an Svideo to COMPOSITE video adapter for about $5US also.
Look [here](RCA female composite video to S-Video male).
In that last link, there were entire cable kits that had S-Video+3.5mm Audio to Composite (Yellow) Video + Stereo Audio (Red_White).
Be careful that the S-VIDEO plug is the same as the output on your Video Card. Sometimes the video cards have the same size & shape plug, but very different plug pin configurations.