Let [tex]x[/tex] be the number of lower-level tickets sold, and [tex]y[/tex] the number of upper-level tickets sold.
Now, if the theater sells 350 tickets total, that means the number of lower-level and upper-level tickets that got sold should add up to 350, so one equation in the system is
[tex]x+y=350[/tex]
You also know that the theater earns $10250 in sales, so if each lower-level ticket costs $35, and each upper-level ticket costs $25[/tex], this means [tex]35x[/tex] represents the amount of money made from selling lower-level ticekts, and [tex]25y[/tex] represents the amount of money made from selling upper-level tickets. You get the second equation in the system,
[tex]35x+25y=10250[/tex]
So the system is
[tex]\begin{cases}x+y=350\\35x+25y=10250\end{cases}[/tex]
Any one of the methods listed in the second part works in solving this system (though I'm not sure what "variable" refers to), but I think substituting involves the least work.
Solve the first equation for either variable; I'll choose [tex]x[/tex].
[tex]x+y=350\implies x=350-y[/tex]
Now substitute this into the second equation for [tex]x[/tex], then solve for [tex]y[/tex].
[tex]35(350-y)+25y=10250\implies 10y=2000\implies y=200[/tex]
So 200 upper-level seats were sold. Meanwhile, since [tex]x+y=350[/tex], this means
[tex]x+200=350\implies x=150[/tex]
so that 150 lower-level tickets were sold.
