To determine if $(3, 7)$ is a solution to the linear system, we substitute $x = 3$ and $y = 7$ into both equations and check if they are satisfied.

1. For the first equation: $y = 2x + 1$ Substitute $x = 3$ and $y = 7$: $7 = 2(3) + 1 \implies 7 = 6 + 1 \implies 7 = 7$ This equation is satisfied.

2. For the second equation: $y = -4x + 7$ Substitute $x = 3$ and $y = 7$: $7 = -4(3) + 7 \implies 7 = -12 + 7 \implies 7 = -5$ This equation is not satisfied.

Since $(3, 7)$ does not satisfy the second equation, it is not a solution to the linear system.