Final answer:
Option D. To find a line parallel to y=x+1 passing through (3,1), use the point-slope formula. The slope must be the same as the original line's slope, which is 1. The correct equation obtained is y=x-2, which is not listed among the provided choices.
Step-by-step explanation:
To write the equation of a line that is parallel to the given line y = x + 1 and passes through the point (3,1), we need to ensure that the slope of our new line is the same as the slope of the given line. Since the given line has a slope of 1 (as we can tell from the coefficient of x in the equation), any line that is parallel to it will also have a slope of 1. Now we'll use the point-slope form of the line equation, which is y - y1 = m(x - x1), where m is the slope and (x1, y1) is a known point on the line, to determine the correct equation.
Substituting the given point and the slope into the point-slope form, we get y - 1 = 1(x - 3). Simplifying this gives us y - 1 = x - 3, and then adding 1 to both sides gives us y = x - 2. This is the equation of the line that is parallel to y = x + 1 and passes through the point (3,1).
Examining the answer choices provided, none of them match our calculated equation, y = x - 2. Thus, there may have been a typo in the question or the answer choices since none of them reflect the correct equation based on the given conditions.