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line e: y =-1/3x+6

line f: y=4x-7
the solution to the system is (3,5)
it is impossible for y to equal both 4x-7 and -1/3x+6 simultaneously; therefore, the system has no solution.
The system is consistent because the lines are perpendicular.
The system is consistent because the lines are NOT parallel.
the system is inconsistent because the lines e and f intersect

1 answer

Let's analyze the information regarding the lines \( e \) and \( f \).

  1. Equations of the lines:

    • Line \( e: y = -\frac{1}{3}x + 6 \)
    • Line \( f: y = 4x - 7 \)

  2. Given Solution:

    • The solution to the system is provided as \( (3, 5) \).

  3. Checking the solution: For point \( (3, 5) \):

    • Substitute \( x = 3 \) into line \( e \): \[ y = -\frac{1}{3}(3) + 6 = -1 + 6 = 5 \]
    • Substitute \( x = 3 \) into line \( f \): \[ y = 4(3) - 7 = 12 - 7 = 5 \] Since both lines yield \( y = 5 \) when \( x = 3 \), the point \( (3, 5) \) is indeed a solution to both equations.

  4. Nature of the lines:

    • To determine if the lines are parallel or intersecting:
      • The slope of line \( e \) is \( -\frac{1}{3} \).
      • The slope of line \( f \) is \( 4 \).
    • Since the slopes are different, the lines are not parallel and therefore must intersect.

  5. Conclusion about the system: Since the two lines intersect at the point \( (3, 5) \), the system of equations is consistent and has exactly one solution.

Summary:

From the statements provided:

  • The system is consistent because the lines are NOT parallel.
  • The statement claiming "it is impossible for \( y \) to equal both" is inaccurate since at \( (3, 5) \) they do equal each other.
  • The statement "the system is inconsistent because the lines e and f intersect" is incorrect, as they are consistent due to the intersection.

Correct Answer:

The system is consistent because the lines are NOT parallel, and they intersect at the point \( (3, 5) \).