As shown in class, the Euclidean algorithm can be used to find solutions to equations of the form
\[ax + by = c.\]

Use the Euclidean algorithm to find integers $x$ and $y$ such that $4x + 3y = 1,$ with the smallest possible positive value of $x$.

Note that while there are many pairs of integers $x$ and $y$ that satisfy this equation, there is only one pair that comes from using the Euclidean algorithm as described in class, and this pair solves the problem.