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 need to check for parallel lines, which occurs when the equations have the same slope but different y-intercepts.

Let's analyze each system:

1. **System 1:**
\[
\begin{cases}
4x + y = 17 \\
4x - y = 3
\end{cases}
\]
- Rearranging the equations:
- \(y = -4x + 17\) (first equation)
- \(y = 4x - 3\) (second equation)
- The slopes are \(-4\) and \(4\) (not parallel, so this system has a solution).

2. **System 2:**
\[
\begin{cases}
4x - y = 17 \\
4x - y = 3
\end{cases}
\]
- Both equations are identical in form, the first can be represented as \(y = 4x - 17\) and the second as \(y = 4x - 3\).
- They have the same slope (\(4\)) but different y-intercepts (17 vs. 3), so these lines are parallel (no solution).

3. **System 3:**
\[
\begin{cases}
4x - y = 17 \\
4x + y = 3
\end{cases}
\]
- Rearranged forms:
- \(y = 4x - 17\)
- \(y = -4x + 3\)
- The slopes are \(4\) and \(-4\) (not parallel), so this system has a solution.

4. **System 4:**
\[
\begin{cases}
4x - y = -17 \\
4x + y = 3
\end{cases}
\]
- Rearranged forms:
- \(y = 4x + 17\)
- \(y = -4x + 3\)
- The slopes are \(4\) and \(-4\) (not parallel), so this system has a solution.

The only system with no solution is **System 2**:

\[
\begin{cases}
4x - y = 17 \\
4x - y = 3
\end{cases}
\]