A linear equation is graphed on the coordinate grid below.
The graph shows a coordinate grid with a straight line that passes through the points (negative 9, negative 5), (negative 6, negative 4) (negative 3, negative 3), (0, negative 2), (3, negative 1), (6, 0), and (9, 1).
When graphed on the same coordinate grid, which equation results in a system of equations with exactly one solution?
A. -2x+6y=-12
B. 3x-9y=9
C. x-3y=6
D.9x-3y=6
To determine which equation results in a system of equations with exactly one solution, we need to look for an equation that represents a line that does not intersect with any of the other lines represented by the given points.
Let's analyze the given points to determine the equation of the line that passes through them:
Point 1: (-9, -5)
Point 2: (-6, -4)
Point 3: (-3, -3)
Point 4: (0, -2)
Point 5: (3, -1)
Point 6: (6, 0)
Point 7: (9, 1)
From the given points, we can observe a pattern: as the x-coordinate increases by 3, the y-coordinate increases by 1. This indicates that the slope of the line is 1/3.
We can use the point-slope form of a linear equation, y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line, to find the equation of the line:
Using Point 1 (-9, -5):
y - (-5) = (1/3)(x - (-9))
y + 5 = (1/3)(x + 9)
3y + 15 = x + 9
x - 3y = -6
Comparing the equation of the line to the given options, we see that the equation x - 3y = -6 is equivalent to option C.
Therefore, the equation x - 3y = 6 results in a system of equations with exactly one solution.