If energy of a system must be conserved, why is it that work can be done on a system?
We must refine the statement "the energy of a system must be conserved" to the more specific and accurate statement "the energy of a system which has zero net force acting on it must be conserved". This is the case because non-zero net force leads to work being performed, either on or by the system. Hence, as I think you have discerned, work on or by a system means a change in that system's energy. In this context, one can see that it is very important to consider what we are setting our system as. You might hear people say "energy of the universe is conserved", which makes sense since we assume that all the forces we know of are acting within, not on, the universe. Therefore zero forces on the universe means net zero force on the universe, hence no work, hence conserved energy.
Can you explain the equation ΔG = ΔH - TΔS and why it is useful?
Absolutely. First, lets think about what ΔG, ΔH, and ΔS mean. ΔG represents the change in Gibbs free energy from the start to the end of some process, like a reaction. Gibbs free energy represents the "ability to do thermodynamic work", but more importantly for us, the change in free energy tells us whether a process is spontaneous or not. A positive value of ΔG means the process is not spontaneous, while a negative value of ΔG means the process is in fact spontaneous. ΔH represents the change in enthalpy, or in other words how much heat is added or released. A negative value of delta H indicates heat is released (exothermic), while a positive value indicates heat is added (endothermic). ΔS represents the change in entropy, or in other words the change in randomness of the system in question. A positive value represents an increase in entropy, while a negative value represents a decrease. The equation ΔG=ΔH-TΔS relates these three values together, along with temperature (T). This equation is tremendously useful because it allows us to figure out the ΔG, therefore if a reaction is spontaneous, using ΔH, T, and ΔS. Note how ΔG will be driven negative by a more negative ΔH, indicating the more exothermic a reaction is, the more likely it will be spontaneous. This makes sense, since systems are driven to minimize their energy, which includes enthalpy (internal energy). It also makes sense that ΔG will be driven negative by a more positive ΔS, since increased randomness in a system is favorable. Temperature serves to magnify the effect of ΔS, whether in a favorable or unfavorable direction. It is possible to have a negative ΔG for an endothermic reaction if TΔS is sufficiently positive, therefore -TΔS sufficiently negative, to overcome the ΔH. Likewise, it is possible to have a negative ΔG for a reaction with negative ΔS if ΔH is sufficiently negative to overcome the TΔS factor.
Can you describe the basics of the macroeconomic output vs. income curve and the aggregate expenditures vs output graph?
Sure thing! To start off, for the output vs income curve, we assume the dollar value of an economy's output equals its income. In other words, how much people make equals how much they produce. Therefore, the equation that describes this graph is simply y=x, where y is output and x is income. It helps to plot this out for visualization purposes. For the aggregate expenditures vs output graph, the equation here is y=(mpe)(x) + AutE , where y is aggregate expenditures, mpe is the marginal propensity to expend, x is output, and AutE is autonomous expenditures. As its name describes, mpe describes the likelihood of spending extra earned money. More money in the hands of Americans means they will go out and spend some amount of that extra money. Therefore it makes sense that mpe is the slope of our graph, since earning more income leads to an increase in spending based on the mpe. Autonomous Expenditures, on the other hand, are an economy's flat-rate, basic expenses, which don't change based on an economy's income (at least to a large extent). Think of this as healthcare spending, or perhaps defense spending. Therefore, it makes sense that autonomous expenditures represent the y-intercept of our graph, since even if an economy's theoretical income was 0 (which wouldn't make much sense by the way), it would still have its autonomous expenditures to pay for.