Question

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 of the given systems of linear equations has no solution, we need to analyze each pair of equations for consistency. A system of equations has no solution if the lines represented by the equations are parallel, meaning they have the same slope but different y-intercepts.

1. **System 1:**
\[
\begin{cases}
4x - y = 17 \\
4x - y = 3
\end{cases}
\]
These equations represent parallel lines with the same slope (4) but different y-intercepts (17 and 3). **This system has no solution.**

2. **System 2:**
\[
\begin{cases}
4x - y = -17 \\
4x + y = 3
\end{cases}
\]
These equations have different slopes since the second equation can be rearranged to give \(y = -4x + 3\). They will intersect at one point. **This system has a solution.**

3. **System 3:**
\[
\begin{cases}
4x - y = 17 \\
4x + y = 3
\end{cases}
\]
Similar to System 2, these equations have different slopes, leading to an intersection point. **This one has a solution as well.**

4. **System 4:**
\[
\begin{cases}
4x + y = 17 \\
4x - y = 3
\end{cases}
\]
Again, these will produce different slopes, thus they will intersect at a single point. **This system also has a solution.**

From the analysis above, the only system of linear equations that has no solution is:

**{4x−y=17, 4x−y=3}**