I want these equations to line up so that all variables and operators go straight down. I've tried a few different techniques but haven't been able to get it to work.
Nope:
\begin{align*}
x+y+z=1 \\
x+y+z=\frac{5}{2} \\
x+y+z=5
\end{align*}
Use &=
to signify to align with the equals:
\begin{align*}
x+y+z &= \,1 \\
x+y+z &= \frac{5}{2} \\
x+y+z &= \,5
\end{align*}
You can use these in math mode:
\; - a thick space \: - a medium space \, - a thin space <-- used this here in front of the simple numbers \! - a negative thin space
Source: http://www.emerson.emory.edu/services/latex/latex_119.html
You can reread about align*
environment f.e. here: https://en.wikibooks.org/wiki/LaTeX/Advanced_Mathematics#align_and_align*