To find an equation that results in a system of equations with exactly one solution, the new equation must have a different slope from the linear equation represented by the points given in your description. A system of equations has exactly one solution when the lines intersect at a single point, which occurs when the lines are not parallel (the slopes are different).
To determine the slope of the original line based on the points provided, we can select two points and calculate the slope (m) using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points (0, -2) and (3, -1):
- \( y_2 = -1 \), \( y_1 = -2 \)
- \( x_2 = 3 \), \( x_1 = 0 \)
Now, substituting into the slope formula:
\[ m = \frac{-1 - (-2)}{3 - 0} = \frac{1}{3} \]
Thus, the slope of the given line is \( \frac{1}{3} \).
Now, let's calculate the slopes of the equations provided in each option:
A. \( -2x + 6y = -12 \)
Rearranging to slope-intercept form \( y = mx + b \):
\[ 6y = 2x - 12 \]
\[ y = \frac{1}{3}x - 2 \]
(Slope = \( \frac{1}{3} \))
B. \( 0.3x - 9y = 9 \)
Rearranging:
\[ -9y = -0.3x + 9 \]
\[ y = \frac{0.3}{9}x + 1 = \frac{1}{30}x + 1 \]
(Slope = \( \frac{1}{30} \))
C. \( x - 3y = 6 \)
Rearranging:
\[ -3y = -x + 6 \]
\[ y = \frac{1}{3}x - 2 \]
(Slope = \( \frac{1}{3} \))
D. \( 9x - 3y = 6 \)
Rearranging:
\[ -3y = -9x + 6 \]
\[ y = 3x - 2 \]
(Slope = 3)
Now, reviewing the slopes:
- The slope of A is \( \frac{1}{3} \) (same as original line).
- The slope of B is \( \frac{1}{30} \) (different).
- The slope of C is \( \frac{1}{3} \) (same as original line).
- The slope of D is 3 (different).
Since we need an equation that does not have the same slope as the original, we find that options B and D have different slopes than the original line.
Therefore, either option B or option D would lead to a system with exactly one solution.
Final answer: B. 0.3x - 9y = 9 or D. 9x - 3y = 6 (both are acceptable responses).