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 find the equation of the graphed line.
First, we can find the slope of the line using the two points (-9, -5) and (9, 1).
The slope is given by: slope = (change in y) / (change in x)
Slope = (1 - (-5)) / (9 - (-9))
Slope = 6 / 18
Slope = 1/3
Next, we can use the point-slope form of a linear equation to find the equation of the line using the point (-9, -5):
y - y1 = m(x - x1)
y - (-5) = (1/3)(x - (-9))
y + 5 = (1/3)(x + 9)
3(y + 5) = x + 9
3y + 15 = x + 9
3y - x = -6
So, the equation of the line is 3y - x = -6.
Now, let's check 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
Plug in the values of x and y from one of the given points into each equation:
For (-9, -5):
A. -2(-9) + 6(-5) = -12
B. 3(-9) - 9(-5) = 9
C. (-9) - 3(-5) = 6
D. 9(-9) - 3(-5) = 6
Looking at the answers, we see that only equation C results in the same value on both sides of the equation when (-9, -5) is plugged in:
(-9) - 3(-5) = 6
Therefore, the equation that results in a system of equations with exactly one solution is C. x - 3y = 6.