Simple closed economy problem Consider a simple closed economy model where Y = C + I + G, consumption is given by the function C = 100 + 0.5 (Y - T), investment is I = 50, government purchases are G = 20, and the government’s 
budget is balanced. 

How do I work out what tax is, to work out the amount of autonomous spending? Is there something obvious I'm missing?
Given your equations, you can find

$$Y = C + 50 + 20 \implies Y = C + 70$$

You can then substitute $Y$ for its equal into your other equation

$$C = 100 + 0.5(C + 70 - T)$$

Finally, given a balanced buget, you know that $G = T$, which should be enough for you to solve for $C$.