To solve the system of equations using substitution:

1. Start with the given equations: $x + 4y = 3$ $2x - y = 4$

2. Solve the first equation for $x$: $x = 3 - 4y$

3. Substitute $x = 3 - 4y$ into the second equation: $2(3 - 4y) - y = 4$ $6 - 8y - y = 4$ $6 - 9y = 4$ $-9y = 4 - 6$ $-9y = -2$ $y = \frac{2}{9}$

4. Substitute $y = \frac{2}{9}$ back into $x = 3 - 4y$: $x = 3 - 4\left(\frac{2}{9}\right)$ $x = 3 - \frac{8}{9}$ $x = \frac{27}{9} - \frac{8}{9}$ $x = \frac{19}{9}$

Thus, the solution is $\left(\frac{19}{9}, \frac{2}{9}\right)$, which matches option (b).