To find which equation results in a system of equations with exactly one solution, we need to identify lines that intersect the line corresponding to the points provided.
First, let's analyze the points given:
- (0, 6)
- (0, -2)
- (-3, -3)
- (-6, -4)
- (-9, -5)
The line passes through the points (0, 6) and (0, -2), which are both on the y-axis. This means that the equation of the line is vertical. The line moving vertically through the y-coordinates indicates that x is constant (here, x = 0).
To find which line has exactly one solution with this line, we need to check for the lines represented by the equations provided, particularly looking for lines with a different slope or an x-intercept not equal to 0.
Let's analyze the options:
A. -2 + 6y = -12:
Rearranging gives \( 6y = -10 \) or \( y = \frac{-10}{6} = \frac{-5}{3} \), which is a horizontal line (constant y-value). This line does not intersect the vertical line; hence it has infinitely many solutions on other points down the y-axis.
B. 3x - 9y = 9:
Rearranging gives \( 9y = 3x - 9 \) or \( y = \frac{1}{3}x + 1 \). This is a line with a slope of 1/3, meaning it would intersect the vertical line at one point (x = 0), specifically at y = 1.
C. x - 3y = 6:
Rearranging gives \( 3y = x - 6 \) or \( y = \frac{1}{3}x + 2 \). This line also has a slope of 1/3 and hence would also have exactly one intersection with the vertical line (x=0).
D. 9x - 3y = 6:
Rearranging gives \( 3y = 9x - 6 \) or \( y = 3x - 2 \). The slope here is 3, which means this has a different slope and a varying y-intercept for different corresponding x intercepts. This line would also intersect the vertical line at exactly one point (x=0, resulting in y=-2).
Considering all options, we find that:
- B gives a slope showing it will intersect at (0,1), which gives one solution.
- C also intersects at (0,2) and thus has one point of intersection.
- D intersects at (0,-2) giving it also a single solution.
The final conclusion is options B and C will produce a single solution; however, since we're looking for only one, I'd assert option D produces a simpler relationship without overlapping with previous points defined.
The final options for systems with exactly one solution are:
- Answer D: 9x - 3y = 6 This option will yield one intersection point hence confirming it is a single solution expressly.
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