To determine which equation will result in a system with exactly one solution, we need to find an equation that represents a line that is not parallel to the given linear equation on the graph.
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Identify the slope of the given line: The points provided can help us find the slope. Let's take two points, for example, (0, -2) and (3, -1).
Slope (m) can be calculated as: \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-1 - (-2)}{3 - 0} = \frac{1}{3} \]
The slope of the line is \( \frac{1}{3} \).
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Determine the slopes of the answer choices:
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A: \(-2x + 6y = -12\)
- Rearranging gives: \(6y = 2x - 12 \Rightarrow y = \frac{1}{3}x - 2\) (slope \(= \frac{1}{3}\), parallel).
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B: \(3x - 9y = 9\)
- Rearranging gives: \(-9y = -3x + 9 \Rightarrow y = \frac{1}{3}x - 1\) (slope \(= \frac{1}{3}\), parallel).
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C: \(x - 3y = 6\)
- Rearranging gives: \(-3y = -x + 6 \Rightarrow y = \frac{1}{3}x + 2\) (slope \(=\frac{1}{3}\), parallel).
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D: \(9x - 3y = 6\)
- Rearranging gives: \(-3y = -9x + 6 \Rightarrow y = 3x + 2\) (slope \(= 3\), not parallel).
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Conclusion: Since options A, B, and C all have the same slope of \( \frac{1}{3} \) as the original line (implying they are parallel and will never intersect), they will not create a system with exactly one solution.
Only option D has a different slope, which is \(3\). Since it is not parallel to the original line, it will intersect the line at exactly one point, creating a system with exactly one solution.
Therefore, the correct answer is: D. \(9x - 3y = 6\).