jmc

We make the variable $x$ the number of adult patrons and $y$ the number of children patrons. Since the movie theater is full and seats $100$, $x+y=100$.

Adult tickets sell for \9.00$each,soatotalof$9x$dollarswerecollectedfromadults.Childrenticketssellfor$\$5.00$ each, so a total of $5y$ dollars were collected from children. A total of \640$werecollected,so$9x+5y=640$.<br/><br/>Wenowhavetwoequations,$x+y=100$and$9x+5y=640$.Wenowsolvefor$y$.<br/><br/>Wemultiplythefirstequationby$9$sothatwecaneliminatethe$x$term:$9x+9y=900$.Then,wesubtractthesecondequationfromittoget$9x+9y-(9x+5y)=900-640 \rightarrow 4y=260 \rightarrow y=65$.<br/><br/>Therefore,$\boxed{65 \text{ children tickets }}$ were sold.