To determine which equation results in a system of equations with exactly one solution when graphed alongside a line that has a positive slope and passes through the points (0, -2), (-6, -4), and (-9, -5), we need to analyze the line and the given equation:
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Finding the equation of the line through the given points:
- The points (0, -2), (-6, -4), and (-9, -5) can be used to determine the slope of the line.
- Let's calculate the slope (m) between two of the points, say (0, -2) and (-6, -4): \[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-4 - (-2)}{-6 - 0} = \frac{-2}{-6} = \frac{1}{3} \]
- Using the slope-intercept form \(y = mx + b\) and the point (0, -2) (where the intersection occurs at the y-intercept \(b\)): \[ y = \frac{1}{3}x - 2 \]
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Converting the line equation to standard form:
- Rearranging it to standard form gives: \[ x - 3y = 6 \ \text{(by multiplying both sides by 3)} \]
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Identifying conditions for a system of equations with one solution:
- For two lines to intersect at exactly one point, they must have different slopes (i.e., they should not be parallel).
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Analyzing the given equation:
- The given equation is: \[ -2x + 6y = -12 \]
- Let's rewrite it in slope-intercept form \(y = mx + b\): \[ 6y = 2x - 12 \implies y = \frac{1}{3}x - 2 \]
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Evaluating the slopes:
- The slope of the equation we calculated from the given points is \(\frac{1}{3}\), which is the same as the slope from the second equation. Therefore, these two equations are actually the same line.
Since they are the same line, they will not intersect at just one point, hence we cannot achieve one unique solution.
In conclusion:
- If you are looking for a different line (with a different slope) to get exactly one solution, you need to find an equation that has a slope that is not \(\frac{1}{3}\). In this case, the option provided is not suitable since it is identical to the original line.
Summary: The equation given does not provide a system with exactly one solution since it represents the same line as the one defined by the points (0, -2), (-6, -4), and (-9, -5). To find an alternative, you would need a line with a different slope.