To determine which equation will result in a system of equations with exactly one solution, we need to identify a line that intersects the given line represented by the points provided.
From the points you provided, we can first find the slope of the line. For example, between the points (-6, -4) and (-3, -3):
\[ \text{slope} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - (-4)}{-3 - (-6)} = \frac{1}{3} \]
Now using point-slope form of the line equation \(y - y_1 = m(x - x_1)\), we can use one of the points, say (0, -2):
\[ y - (-2) = \frac{1}{3}(x - 0) \] \[ y + 2 = \frac{1}{3}x \] \[ y = \frac{1}{3}x - 2 \]
This means the line can be expressed as \(y = \frac{1}{3}x - 2\).
For a new equation (option A, B, C, or D) to intersect this line at exactly one point, it must have a different slope. If two lines have the same slope, they will be parallel and never intersect (no solution), or they could be the same line (infinitely many solutions).
Now, you want to choose one option that has a different slope than \(\frac{1}{3}\). Possible slopes could be any value that is not equal to \(\frac{1}{3}\).
Please provide the equations for options A, B, C, and D, so we can determine which has a different slope for exactly one intersection.
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