Question

Which of the following systems of linear equations has no solution?(1 point)
Responses

{4x−y=−174x+y=3

{4x−y=174x+y=3

{4x+y=174x−y=3

{4x−y=174x−y=3

Answer 1

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:

\[
\{4x - y = 17, \, 4x - y = 3\}
\]