System of Equations Algebraic Equations Word Problem The problem is
"In a math class that contains both 7th and 8th graders, each student must do a class presentation on a famous mathematician. Each student may do the presentation alone or with a class partner. A 7th grader's partner must be an 8th grader and an 8th grader's partner must be a 7th grader. If two-thirds of the 7th graders and three-fiths of the 8th graders work with partners, what fraction of the class works alone? Solve using a system of equations."
I've come up with the equations $\frac{2}{3}x = \frac{3}{5}y$ and $z = \frac{1}{3}x + \frac{2}{5}y$ but I'm not entirely sure whether these are correct/where to go form here.

You are halfway there!  
The fraction of students working alone is $$q= \frac{z}{z+\frac{2}{3} x+ \frac{3}{5}y}$$
This is $$q=\frac{\frac{1}{3} x + \frac{2}{5} y }{x+y}$$
Your first equation implies $x=\frac{9}{10} y$, so
$$ q = \frac{\left(\frac{9}{10}\right)\left(\frac{1}{3}\right) + \frac{2}{5} }{\frac{19}{10}}= \frac{7}{19}.$$
