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-9x =9
C. x-3y=6
D. 9x-3y =6
To determine which equation will result in a system of equations with exactly one solution, we need to find the equation of the line that passes through the given points. Using any two of the given points, we can determine the slope of the line using the formula: slope = (change in y) / (change in x).
Using the points (-9,-5) and (-6,-4):
slope = (-4 - (-5)) / (-6 - (-9))
= (-4 + 5) / (-6 + 9)
= 1 / 3
Using the points (0,-2) and (3,-1):
slope = (-1 - (-2)) / (3 - 0)
= (-1 + 2) / (3 - 0)
= 1 / 3
Since the slope is the same for all the given points, it means that the line is a straight line and can be described by the equation y = mx + b, where m is the slope and b is the y-intercept.
Using the point (-9,-5):
-5 = (1/3)(-9) + b
-5 = -3 + b
b = -5 + 3
b = -2
So the equation of the line is y = (1/3)x - 2.
Now, we can compare this equation to the answer choices:
A. -2x + 6y = -12
Converting to slope-intercept form, we get:
6y = 2x - 12
y = (2/6)x - 2
This equation has the same slope but different y-intercept. The system of equations will have a unique solution.
B. 3x - 9x = 9
This equation simplifies to:
-6x = 9
x = -9/6
x = -3/2
The equation is not in the form y = mx + b and does not represent a line. Therefore, it will not have exactly one solution.
C. x - 3y = 6
Converting to slope-intercept form, we get:
-3y = -x + 6
y = (1/3)x - 2
This equation has the same slope and y-intercept as the given line. The system of equations will have infinitely many solutions.
D. 9x - 3y = 6
Converting to slope-intercept form, we get:
-3y = -9x + 6
y = 3x - 2
This equation has the same slope but different y-intercept. The system of equations will have a unique solution.
Therefore, the answer is option A. -2x + 6y = -12.