Sorry, I probably went too fast. This is a tricky word problem.

So with these word problems, the first thing you always need to do is know what you’re solving for. In this case, we are solving for: “How much should they charge per person given that the band wants to earn $67300?” and “How many people will attend the show given how much the band will charge?”

So we should define the dependent variable p = \{\text{how many people attended the show}\} and the independent variable c = \{\text{how much the band charges per person} \}. Additionally, because c influences p (if c decreases by 2 dollars, then p increases by 150 people), we can write an equation that shows this relationship between the two.

We know that p and c are two quantities that are related by a linear equation (like y=mx+b, where y is p as the independent variable, and x is c as the dependent variable) precisely because when c changes a fixed amount (-2), p also changes a fixed amount (150).

So to find the line that this relationship represents we should use the point-slope formula, where p - p_0 = m(c - c_0), where m is the slope and (c_0,p_0) is a point on the line.

Well, so what is m, the slope? It’s what we talked about earlier: whenever c decreases by 2, then p increases by 150, so the slope must be \frac{150}{-2} = -75.

So now our equation looks like p - p_0 = -75(c - c_0). We’re almost done with finding the equation, but we need to know what the point (c_0,p_0) is. Luckily, at the beginning of the problem, we’re given that the band Gov’t Mule the year before charged $100 per person and 3000 people attended the show, so we know that the point (c_0,p_0) = (100, 3000) has to be on the line.

So now our equation looks like p - 3000 = -75(c - 100). And if we simplify it, it looks like p = -75c + 10500. Great, so now we know how the number of people who attend changes as the price per person changes.

So, just as a quick example, if the band chose to charge 50 dollars per person, then you just substitute 50 for c to get that p = -75(50) + 10500 = 6750 people attended.

So yeah, we have the equation that relates p and c. But we’re not done, because we haven’t answered the question, given that the band only wants to earn $67,300, how much should they charge, and how many people attend if they charge that amount?

So if the band only wants $67,300, then it must be the case that pc = 67300, since pc is the total revenue that the band makes from ticket sales. Think about how if, for example, the concert had 10 people and it was $2 per ticket, the revenue would be 10\cdot 2 = 20 dollars.

So now we have two equations with two unknowns, so we can solve it:

\begin{cases} p = -75c + 10500 ~~~~~(1)\\ pc = 67300 ~~~~~~~~~~~~~~~~~(2) \end{cases}\tag*{}

Take equation (2) and solve for p to get p = \dfrac{67300}{c} and plug this into (1) and solve:

\dfrac{67300}{c} = -75c + 10500\tag*{}

67300 = -75c^2 + 10500c \tag*{}

0 = 75c^2 - 10500c + 67300 \tag*{}

c = 6.773 \text{ or } 133.3\tag*{}

Plug these into either of the equations (1) or (2) to get the corresponding values of p: p = 9950, 505.

Since we want the solution with the largest population, we should choose the solution (6.773, 9950). And so we’re done.

Let me know if you have any other specific questions!