line e: 2x-4y=0

Question

line f: 2x-4y=12Which statements about the system are true? (Choose two.)

Responses
The solutions to the system are (0, 0) and (0, –3).
it is impossible for 2x-4y to equal both 0 and 12 simultaneously; therefore, the system has no solution.
It is impossible for 2 x minus 4 y to equal both 0 and 12 simultaneously; therefore, the system has no solution.
The system is inconsistent because the lines are parallel.
The system is inconsistent because the lines are parallel.
The system is consistent because the lines are parallel.
The system is consistent because the lines are parallel.
The system has no solution because the lines are perpendicular.

Answer 1

To analyze the situation with the given lines:

1. Line e: \(2x - 4y = 0\) can be rewritten as \(y = \frac{1}{2}x\) (i.e., it has a slope of \( \frac{1}{2} \)).
2. Line f: \(2x - 4y = 12\) can be rewritten as \(y = \frac{1}{2}x - 3\) (i.e., it also has a slope of \( \frac{1}{2} \)).

Since both lines have the same slope but different y-intercepts, they are parallel and will never intersect.

Now, considering the statements provided:

1. It is impossible for \(2x - 4y\) to equal both 0 and 12 simultaneously; therefore, the system has no solution. - This statement is true.

2. The system is inconsistent because the lines are parallel. - This statement is also true.

Based on this analysis, the two correct responses are:

• It is impossible for \(2x - 4y\) to equal both 0 and 12 simultaneously; therefore, the system has no solution.
• The system is inconsistent because the lines are parallel.