To find the equation of the line that passes through the points given by the x and y values, we first determine the slope (m) using the points (let's use the first two points: (0, 8) and (3, 6)):
The slope formula is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Using the points (0, 8) and (3, 6): \[ m = \frac{6 - 8}{3 - 0} = \frac{-2}{3} \]
Now we have the slope \(m = -\frac{2}{3}\).
Next, we can use the point-slope form of the equation of a line, which is: \[ y - y_1 = m(x - x_1) \] Using the point (3, 6): \[ y - 6 = -\frac{2}{3}(x - 3) \]
This matches option A perfectly: A: \(y - 6 = -\frac{2}{3}(x - 3)\)
Next, let's examine the other options to see if they represent the same line.
Option B: \(3y = -2x + 24\) Dividing by 3: \[ y = -\frac{2}{3}x + 8 \] This matches the slope-intercept form of the line, so it is equivalent.
Option C: \(2x + 3y = 24\) Rearranging this gives: \[ 3y = -2x + 24 \quad \Rightarrow \quad y = -\frac{2}{3}x + 8 \] This is equivalent to \(y = -\frac{2}{3}x + 8\).
Option D: \(y = -\frac{2}{3}x + 8\) This is also equivalent to the same line.
Thus, all equations represent the same line. However, if you are looking for the correct options selected for the original question:
The answer matches A, but all options given that present the same line are correct, including B, C, and D.