[JaredL]

It's been a couple years since I've done this but hopefully I can provide some info.

Firstly, we're doing constrained optimization. So you are finding the best bundle you can subject to not spending more than your income (y in your setup).

The easiest place to start is (3). The first equation says that for x_i one of two things must be true. Either you purchase none of it or you buy just enough so that the marginal benefit equals the marginal cost. The marginal benefit is the partial derivative of U wrt x_i. The marginal cost is how much of your budget you lose which is p_i (the marginal monetary cost of buying good x_i) times the marginal utility of income. Think of p_i as the amount of money you lose and lambda as the price of that loss.

The reason for this restriction is that the price of good x_i may be such that you would optimally purchase none of the good. Think about how the price of Ferraris affects your purchasing decisions. Similarly, think about how many more ferraris you will buy if you get $1 richer. So in the case of a Ferrari your marginal benefit is probably very high but the price is so high that equation 1 will be strictly negative. Because you won't purchase any, equation 3 is satisfied.

Now look at the second equation in (3). Lambda, which as you said is marginal benefit of income, times the difference between how much money you have and how much you are spending must be zero. This is simply saying that if more money is useful to you (lambda > 0) then you are spending all of your money (y - p*x = 0). Most utility functions are increasing so all money will be spent.

So the punchline here is that you know in the optimal bundle either a positive amount of each good will be consumed, in which case you can solve by setting the marginal utility of x1 divided its price equal to the same for x2 and then plugging that in to p1x1 + p2x2 = y. Or it could be at a corner. The only way to know in general is to check. Some utility functions feature infinite marginal utility for each good at 0 so that the corners are never optimal, but otherwise you just have to check them to make sure that you don't get more utility from a corner than the interior optimum.