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

Question

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 check for parallel lines, which occur when the lines have the same slope but different y-intercepts.

Let's analyze each system:

1. System 1:
   $$\{\begin{array}{l}4x-y=174x-y=3\end{array}$$

Here, both equations have the same left-hand side (same slope) but different right-hand sides (17 and 3), indicating that the lines are parallel and will never intersect. Therefore, this system has no solution.

2. System 2:
   $$\{\begin{array}{l}4x+y=174x-y=3\end{array}$$

The slopes of the first equation is -4 (rearranging gives us $y=-4x+17$) and the second equation has a slope of 4 (rearranging gives us $y=4x-3$). Since the slopes are different, these lines will intersect. Thus, this system has a solution.

3. System 3:
   $$\{\begin{array}{l}4x-y=174x+y=3\end{array}$$

Again, the first equation has a slope of -4, while the second equation has a slope of 4. Since the slopes are different, these lines will also intersect. Thus, this system has a solution.

4. System 4:
   $$\{\begin{array}{l}4x-y=-174x+y=3\end{array}$$

Similar to the previous systems, the first equation has a slope of -4 and the second equation has a slope of 4. These lines will intersect, meaning this system also has a solution.

Based on the analysis, the system that has no solution is:

System 1: $$\{\begin{array}{l}4x-y=174x-y=3\end{array}$$

This system is the only one with parallel lines.