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
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Answer 1

To determine which of the given systems of linear equations has no solution, we need to analyze whether the equations in each system are consistent or inconsistent. A system of linear equations has no solution if the lines represented by the equations are parallel, meaning they have the same slope but different y-intercepts.

Let's examine each option:

1. **System:**
\[
\begin{cases}
4x + y = 17 \\
4x - y = 3
\end{cases}
\]

Rearranging the second equation:
\[
4x - y = 3 \implies y = 4x - 3
\]
The first equation rearranges to:
\[
y = -4x + 17
\]
Here, the slopes are -4 and 4 (+4 for the second equation), hence they are not parallel. This system has a solution.

2. **System:**
\[
\begin{cases}
4x - y = -17 \\
4x + y = 3
\end{cases}
\]
Rearranging the second equation:
\[
y = -4x + 3
\]
The first equation can be rearranged to:
\[
y = 4x + 17
\]
Again, the slopes here are 4 and -4, so they are not parallel. This system has a solution.

3. **System:**
\[
\begin{cases}
4x - y = 17 \\
4x - y = 3
\end{cases}
\]
Both equations describe the same line except for the constant, leading to:
- The first equation: \(4x - y = 17\)
- The second equation: \(4x - y = 3\)
Since they are essentially two equations of the same form but with different constants, the lines are parallel. This system does not have a solution.

4. **System:**
\[
\begin{cases}
4x - y = 17 \\
4x + y = 3
\end{cases}
\]
Rearranging the second equation:
\[
y = -4x + 3
\]
The first becomes:
\[
y = 4x - 17
\]
Again, the slopes are different (4 and -4), indicating that this system has a solution.

Thus, the correct answer is:

**{4x - y = 17, 4x - y = 3** (This system has no solution.)