## Toymaker

A toymaker is making miniature 3-wheel and 4-wheel all-terrain vehicles (ATVs). He has a total of $61$ wheels and $17$ sets of handlebars. How many 4-wheel ATVs can he make?
Source: NCTM Math Teacher 2008

SOLUTION
If the toymaker makes $15$ 4-wheel ATVs, he uses $15\times 4=60$ wheels and $15$ sets of handlebars. The remaining $1$ wheel and $2$ handlebars are not enough to make any 3-wheel ATVs. If he makes $14$ 4-wheel, he uses $14\times 4=56$ wheels and $14$ handlebars. The remaining $5$ wheels and $3$ handlebars are enough to make $1$ 3-wheel ATVs leaving $2$ wheels and $2$ handlebars unused.
The following table summarizes the different combinations of ATVs
4-wheel    3-wheel    wheels unused    handlebars unused
$15\qquad\quad\:\: 0\qquad\qquad 1\qquad\qquad\qquad\:\:\: 2$
$14\qquad\quad\:\: 1\qquad\qquad 2\qquad\qquad\qquad\:\:\: 2$
$13\qquad\quad\:\: 3\qquad\qquad 0\qquad\qquad\qquad\:\:\: 1$
$12\qquad\quad\:\: 4\qquad\qquad 1\qquad\qquad\qquad\:\:\: 1$
$11\qquad\quad\:\: 5\qquad\qquad 2\qquad\qquad\qquad\:\:\: 1$
$10\qquad\quad\:\: 7\qquad\qquad 0\qquad\qquad\qquad\:\:\: 0$
$9\qquad\quad\:\:\:\: 8\qquad\qquad 1\qquad\qquad\qquad\:\:\: 0$
$8\qquad\quad\:\:\:\: 9\qquad\qquad 2\qquad\qquad\qquad\:\:\: 0$
Two possible solutions: $10$ 4-wheel and $7$ 3-wheel or $13$ 4-wheel and $3$ 3-wheel. Both solutions use all of $61$ wheels but the first one is  better because it uses all of 17 sets of handlebars.

Alternative solution
The following figure shows the graph of the Diophantine equation $4x+3y=61$

Two points $(10,7)$ and $(13,3)$ on the graph are lattice points.

Answer: $10$