John has a lemonade stand. He sells a small lemonade for $50$ cents and a large lemonade for $\1$. A small serving contains $1$ cup of lemonade; a large contains $1.5$ cups. At the end of the day, John has made $\9$ and sold $15.5$ cups of lemonade. How many small and large lemonades has he sold?
Source: NCTM Mathematics Teacher, September 2006

Solution
Let $x$ represent the number of small lemonades and $y$ the number of large lemonades sold. We have the following system of equations
$x(.50)+y(1)=9\qquad\qquad\; (1)$
$x(1)+y(1.5)=15.5\qquad\quad\, (2)$
Multiply Eqs. $(1)$ and $(2)$ by $2$
$x+2y=18\qquad\quad\:\: (3)$
$2x+3y=31\qquad\quad (4)$
Multiply Eq. $(3)$ by $-2$ and add to Eq. $(4)$
$-2x-4y=-36$
$2x+3y=31$
————————-
$-y=-5$
$y=5$
Substitute the value of $y=5$ into Eq. $(3)$
$x+2(5)=18$
$x=8$
John has sold $8$ small and $5$ large lemonades.

Answer: $8$ small and $5$ large