To determine which system of linear equations has no solution, we look for systems that represent parallel lines. Parallel lines have the same slope but different y-intercepts.
Let's analyze each system:
1. **System:**
- \(4x - y = -17\)
- \(4x + y = 3\)
- Convert to slope-intercept form \(y = mx + b\):
- First equation: \(y = 4x + 17\) (slope \(m = 4\))
- Second equation: \(y = -4x + 3\) (slope \(m = -4\))
- **Conclusion:** Different slopes → has a solution (intersecting lines).
2. **System:**
- \(4x - y = 17\)
- \(4x + y = 3\)
- Convert to slope-intercept form:
- First equation: \(y = 4x - 17\) (slope \(m = 4\))
- Second equation: \(y = -4x + 3\) (slope \(m = -4\))
- **Conclusion:** Different slopes → has a solution (intersecting lines).
3. **System:**
- \(4x + y = 17\)
- \(4x - y = 3\)
- Convert to slope-intercept form:
- First equation: \(y = -4x + 17\) (slope \(m = -4\))
- Second equation: \(y = 4x - 3\) (slope \(m = 4\))
- **Conclusion:** Different slopes → has a solution (intersecting lines).
4. **System:**
- \(4x - y = 17\)
- \(4x - y = 3\)
- The first equation can be rewritten as \(y = 4x - 17\) and the second equation as \(y = 4x - 3\). Both equations have the same slope (\(m = 4\)) but different y-intercepts.
- **Conclusion:** The lines are parallel and never intersect → this system has no solution.
From the analysis, the system of linear equations that has no solution is:
\[
\{4x - y = 17, \, 4x - y = 3\}
\]