To analyze the system of equations you provided,
- Line e: \( y = -\frac{1}{3}x + 6 \)
- Line f: \( y = 4x - 7 \)
You mentioned that the solution to the system is \( (3, 5) \). Let's verify this solution:
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For line e: \[ y = -\frac{1}{3}(3) + 6 = -1 + 6 = 5 \] So, the point \( (3, 5) \) satisfies line e.
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For line f: \[ y = 4(3) - 7 = 12 - 7 = 5 \] So, the point \( (3, 5) \) satisfies line f as well.
Since \( (3, 5) \) satisfies both equations, it confirms that the system has a solution.
Now, let's evaluate the statements made:
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"it is impossible for y to equal both 4x-7 and -1/3x+6 simultaneously; therefore, the system has no solution."
- This is false. We found that \( (3, 5) \) is a solution.
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"The system is consistent because the lines are perpendicular."
- This is not entirely accurate. Although the lines are perpendicular (the slopes \(-\frac{1}{3}\) and \(4\) yield a product of \(-1\)), the key point is that they intersect at \( (3, 5) \).
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"The system is consistent because the lines are NOT parallel."
- This is true. The lines are not parallel, which means they will intersect at some point, hence the system is consistent.
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"the system is inconsistent because the lines e and f intersect."
- This is contradictory. The fact that the lines intersect means that the system is consistent, not inconsistent.
In summary, the correct statement about the system is that it is consistent because the lines are not parallel (they intersect at \( (3, 5) \)). Also, they are perpendicular, which is an additional valid statement about their relationship.
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