To determine which equation will result in a system of equations with exactly one solution when graphed, we must understand that two linear equations will intersect at exactly one point if they are not parallel.
The equations are considered parallel if they have the same slope. We can first determine the slope of the given equation \(-2x + 6y = -12\):
- Rearranging it to slope-intercept form (y = mx + b): \[ 6y = 2x - 12 \] \[ y = \frac{1}{3}x - 2 \] Here, the slope (\(m\)) is \(\frac{1}{3}\).
Next, we will find the slope of each of the potential solutions to see if they share the same slope (which means they will be parallel), or if they have different slopes (which means they will intersect at a single point).
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Equation 1: \(3x - 9y = 9\) \[ -9y = -3x + 9 \] \[ y = \frac{1}{3}x - 1 \] Slope: \(\frac{1}{3}\) (parallel)
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Equation 2: \(x - 3y = 6\) \[ -3y = -x + 6 \] \[ y = \frac{1}{3}x + 2 \] Slope: \(\frac{1}{3}\) (parallel)
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Equation 3: \(9x - 3y = 6\) \[ -3y = -9x + 6 \] \[ y = 3x - 2 \] Slope: \(3\) (not parallel)
Therefore, the equation that when graphed will result in a system of equations with exactly one solution is \(9x - 3y = 6\), as it has a different slope than \(-2x + 6y = -12\).
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