Final answer:
The equation of the line through the points (3,6) and (5,-8) is found by first calculating the slope, which is -7. Plugging this slope into the point-slope form with one of the points, we derive the equation y = -7x + 27. However, none of the provided options match this equation, suggesting a potential error in the question.
Step-by-step explanation:
The question is asking for the point-slope equation of the line passing through the two points (3,6) and (5,-8). To find this, we first calculate the slope of the line, which is the change in y divided by the change in x:
Slope (m) = (y2 - y1) / (x2 - x1) = (-8 - 6) / (5 - 3) = -14 / 2 = -7.
Now we can use the point-slope form of the equation, y - y1 = m(x - x1), using the slope and one of the points (3,6). Substituting these values in gives us:
y - 6 = -7(x - 3).
Expanding this, we get:
y - 6 = -7x + 21.
Adding 6 to both sides of the equation, we have:
y = -7x + 27.
Thus, the completed point-slope equation that fits the provided options is closest to y + 7x = 9, though none of the options exactly matches our derived equation. If we simplify our derived equation by subtracting 7x from both sides, we get:
-7x + y = 27, which when simplified further by dividing every term by -1 gives us:
7x - y = -27.
This is not listed as an option, so it seems there might be an error in the given choices or in the instruction. It is crucial to double-check the instruction to ensure it matches the calculated equation.