My daughter got the following optional question for 8th grade homework mixed into her simultaneous equations questions. All of the other questions were simple for me to solve using substitution or elimination which is what she is being taught at the moment but this one seems to me to be at a much higher grade.
A high school play sold 168 tickets to one showing. If tickets were bought in advance, they were half price. If they made $2130 from that one show, how many tickets were sold in advance?
I put together a few formulas but had to use guesswork for the last step which just felt wrong because math should be about processes, not guesswork.
a = advance r = regular v = advance price
$$a + r = 168$$ $$a(v) + r(2v) = 2130$$ $$a > 0$$ $$r > 0$$
Solving I get:
$$v(168-r) + 2rv = 2130$$ $$168v - rv + 2rv = 2130$$ $$168v + rv = 2130$$ $$v(168 + r) = 2130$$
At this point I guessed v at 10 and worked out r and a from that. This concerned me because I can't teach my daughter to make educated guesses. What is the process/trick involved with solving these sort of questions?