what is a statement that best describes the system of equations {y = -3x + 7, y = -3x - 7?

Answer 1

The system of equations \( y = -3x + 7 \) and \( y = -3x - 7 \) represents two parallel lines. Since they both have the same slope of \(-3\) but different y-intercepts (\(7\) and \(-7\)), they will never intersect, meaning there is no solution to the system. Therefore, the statement that best describes this system is:

"The system has no solutions, as the equations represent two parallel lines."

Answer 2

which of following systems of linear equations has no solution
1. {4x + y = 17, 4x - y = 3
2. {4x - y = 17, 4x + y = 3
3. {4x - y = 17, 4x - y = 3
4. {4x - y = -17, 4x + y = 3

Answer 3

To determine which of these systems of linear equations has no solution, we need to check for systems that represent parallel lines. Lines are parallel if they have the same slope but different y-intercepts.

1. For the system {4x + y = 17, 4x - y = 3:

   • First equation: \(y = -4x + 17\) (slope = -4)
   • Second equation: \(y = 4x - 3\) (slope = 4)
   • Different slopes → This system has a solution.

2. For the system {4x - y = 17, 4x + y = 3:

   • First equation: \(y = 4x - 17\) (slope = 4)
   • Second equation: \(y = -4x + 3\) (slope = -4)
   • Different slopes → This system has a solution.

3. For the system {4x - y = 17, 4x - y = 3:

   • Both equations represent the same line rearranged, which gives:
     • First equation: \(4x - y = 17\)
     • Second equation: \(4x - y = 3\)
   • These are essentially the same lines, providing conflicting results. Therefore, this system has no solution.

4. For the system {4x - y = -17, 4x + y = 3:

   • First equation: \(y = 4x + 17\) (slope = 4)
   • Second equation: \(y = -4x + 3\) (slope = -4)
   • Different slopes → This system has a solution.

Thus, the system of linear equations that has no solution is option 3: {4x - y = 17, 4x - y = 3}.