Final answer:
Both equations A (y = 2/3x + 5) and B (y - 1 = 2/3(x + 6)) represent a line that passes through the points (-6,1) and (9,11) after calculating the slope and using point-slope form. Hence, the correct answer is D, both I and II.
Step-by-step explanation:
To determine which equation represents a line that passes through the points (-6,1) and (9,11), we first calculate the slope of the line using the formula: slope (m) = (y2 - y1) / (x2 - x1). Plugging in the points, we get m = (11 - 1) / (9 - (-6)) = 10 / 15 = 2 / 3.
Next, we use the point-slope form of a linear equation, y - y1 = m(x - x1), with either of the points. Using the point (-6,1), we write the equation as y - 1 = (2 / 3)(x + 6), which matches option B.
We can also convert this point-slope form into the slope-intercept form, y = mx + b, to see if it matches option A. Distributing 2 / 3 into (x + 6) and adding 1 to both sides, we obtain y = (2 / 3)x + 4 + 1 = (2 / 3)x + 5. This is exactly what option A presents.
Thus, both equations A and B represent the same line that passes through the given points. The answer to the question is therefore D, both I and II.