And let s be the cost of one shirt and t be the cost of one tie.
So S*s + T*t = 10 000.
3s = 100, t = 20.
So if we rewrite 3s = 100 as s = 100/3, and filling this and t = 20 into the first equation we have:
100/3 * S + 20*T = 10 000.
Now, if we sold half the shirts and two-thirds of the ties is if we sell 1/2 * S and 2/3 * T, and from selling this we get 6000.
So 1/2 * S * s + 2/3 * T * t = 6000.
or, filling in s = 100/3 and t = 20 as before:
1/2 * S * 100/3 + 2/3 * T * 20 = 6000.
So there you have your two simultaneous equations:
100/3 * S + 20*T = 10000
100/6 * S + 40/3 * T = 6000
Solving for S and T will give you the total amount of shirts and ties.
And the question is how many of each did he sell in his sale. Well, he sold all of them. So S is the number of shirts he sold, and T is the number of ties he sold...
(And just to check with your given answers so as I don't get it wrong and look a complete fool:
100/3 * 120 + 20*300 = 4000 + 6000 = 10000.
100/6*120 + 40/3*300 = 2000 + 4000 = 6000.
So yes - your method was fine!
You got your simultaneous equations, and you just had to finish it out...
( Note to self: read entire post in future...)