To find an equation that results in a system of equations with exactly one solution, we need to identify a line that has a different slope than the line represented by the points provided. The points you mentioned create a linear pattern, and we can determine the slope of that line.
Let's find the slope using two points from the list, for example, (0, -2) and (3, -1):
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Calculate the change in y (Δy) and change in x (Δx):
- Δy = -1 - (-2) = 1
- Δx = 3 - 0 = 3
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The slope (m) is calculated as:
- m = Δy / Δx = 1 / 3
Thus, the slope of the line represented by the points is \( \frac{1}{3} \).
For a system of equations to have exactly one solution, the new line must have a different slope than \( \frac{1}{3} \).
Now let's check the slope of each of the given equations to see which one has a different slope:
A. -2x + 6y = -2
Rearranging gives:
\[ 6y = 2x - 2 \quad \Rightarrow \quad y = \frac{1}{3}x - \frac{1}{3} \]
(Slope is \( \frac{1}{3} \))
B. 3x - 9y = 9
Rearranging gives:
\[ -9y = -3x + 9 \quad \Rightarrow \quad y = \frac{1}{3}x + 1 \]
(Slope is \( \frac{1}{3} \))
C. x - 3y = 6
Rearranging gives:
\[ -3y = -x + 6 \quad \Rightarrow \quad y = \frac{1}{3}x - 2 \]
(Slope is \( \frac{1}{3} \))
D. 9x - 3y = 6
Rearranging gives:
\[ -3y = -9x + 6 \quad \Rightarrow \quad y = 3x - 2 \]
(Slope is \( 3 \))
From the analysis, we see that option D has a slope of 3, which is different from \( \frac{1}{3} \). Thus, it will intersect with the original line at exactly one point.
Therefore, the answer is:
D. 9x - 3y = 6