To determine which system of equations has no solution, we need to check if the equations in each system are inconsistent, meaning they represent parallel lines that never intersect.
1. **First System:**
\[
\begin{align*}
4x - y &= -17 \quad (1) \\
4x + y &= 3 \quad (2)
\end{align*}
\]
Adding equations \( (1) \) and \( (2) \):
\[
(4x - y) + (4x + y) = -17 + 3 \implies 8x = -14 \implies x = -\frac{14}{8} = -\frac{7}{4}
\]
Substitute \( x = -\frac{7}{4} \) back into either equation to solve for \( y \):
\[
4(-\frac{7}{4}) - y = -17 \implies -7 - y = -17 \implies y = 10
\]
This system has a solution.
2. **Second System:**
\[
\begin{align*}
4x - y &= 17 \quad (1) \\
4x + y &= 3 \quad (2)
\end{align*}
\]
Adding equations \( (1) \) and \( (2) \):
\[
(4x - y) + (4x + y) = 17 + 3 \implies 8x = 20 \implies x = 2.5
\]
Substitute \( x = 2.5 \) back into either equation to solve for \( y \):
Equation (1):
\[
4(2.5) - y = 17 \implies 10 - y = 17 \implies y = -7
\]
This system has a solution.
3. **Third System:**
\[
\begin{align*}
4x + y &= 17 \quad (1) \\
4x - y &= 3 \quad (2)
\end{align*}
\]
Adding equations \( (1) + (2) \):
\[
(4x + y) + (4x - y) = 17 + 3 \implies 8x = 20 \implies x = 2.5
\]
Substitute \( x = 2.5 \):
\[
4(2.5) + y = 17 \implies 10 + y = 17 \implies y = 7
\]
This system has a solution.
4. **Fourth System:**
\[
\begin{align*}
4x - y &= 17 \quad (1) \\
4x - y &= 3 \quad (2)
\end{align*}
\]
Since both equations are equivalent (\( 4x - y \) is the same in both equations), they yield:
\[
17 \neq 3
\]
This system is inconsistent and has no solution.
Therefore, the system that has no solution is:
\[
\{4x - y = 17 \\ 4x - y = 3\}
\]