A linear equation is graphed on the coordinate plane 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 plane, 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 will result in a system of equations with exactly one solution when graphed with the given linear equation, we need to look for an equation that has a different slope from the given linear equation.
The given linear equation passes through the points (-9, -5), (-6, -4), (-3, -3), (0, -2), (3, -1), (6, 0), and (9, 1). By observing the points, we can see that the slope of the given linear equation is 1/3.
Now, let's analyze each of the answer choices:
A. -2x + 6y = -12
To find the slope-intercept form of this equation, we solve for y:
6y = 2x - 12
y = 1/3x - 2
The slope of this equation is 1/3, which is the same as the given linear equation. Therefore, this system of equations will have infinitely many solutions.
B. 3x - 9y = 9
To find the slope-intercept form of this equation, we solve for y:
-9y = -3x + 9
y = 1/3x - 1
The slope of this equation is 1/3, which is the same as the given linear equation. Therefore, this system of equations will have infinitely many solutions.
C. x - 3y = 6
To find the slope-intercept form of this equation, we solve for y:
-3y = -x + 6
y = 1/3x - 2
The slope of this equation is 1/3, which is the same as the given linear equation. Therefore, this system of equations will have infinitely many solutions.
D. 9x - 3y = 6
To find the slope-intercept form of this equation, we solve for y:
-3y = -9x + 6
y = 3x - 2
The slope of this equation is 3, which is different from the slope of the given linear equation (1/3). Therefore, this system of equations will have exactly one solution.
Therefore, the equation that results in a system of equations with exactly one solution when graphed with the given linear equation is D. 9x - 3y = 6.