Troubleshooting Utility Maximization with the Lagrange Method I'm trying to solve a utility maximization problem through the Lagrange method. 
The utility function is something like $u(x,y)=x+B(y-a)$.

However, I'm running into problems as x and y do not come up in the First Order Conditions. I was wondering if anyone had an intuitive explanation as for why, and how to get around it?
You needn't use the Lagrange here. And in general, this will be true for additively separable, linear utility functions. 

Note that the marginal rate of substitution between any two goods $i,j$ is a utility function of the form $u(x_1,x_2,...,x_n)=a_1x_1+...+a_nx_n$ will be constant at $MRS_{i,j}=\frac{a_i}{a_j}$

A consumer will buy none of a good $x_i$ whenever $\frac{u_i}{p_i}$ is below the maximum such ratio of the $n$ goods. A consumer will exhaust her budget on whichever good has the maximum such ratio. And if all such ratios are equal, the consumer is indifferent between any consumption bundle that exhuats wealth $w$. 

For your case of two goods, indifference curves are linear. For more than two goods, they are hyperplanes.