Final answer:
The student's question revolves around finding linear equations in two forms and determining the relationship between two lines. The slopes of the lines are calculated and compared to establish if the lines are parallel, perpendicular or neither.
Step-by-step explanation:
The subject of the student's question pertains to determining the equation of a line in both slope-intercept form and point-slope form, as well as analyzing the relationship between two lines to see if they are parallel, perpendicular, or neither. To find the slope of a line passing through two points, we use the slope formula, which is the rise over the run, or the change in y divided by the change in x (m = (y2 - y1) / (x2 - x1)). Using this formula, we can calculate the slopes of the two lines and determine their relationship.
For the first line through points (16, 3) and (1, -1), the slope is (3 - (-1))/(16 - 1) = 4/15. An equation in slope-intercept form (y = mx + b) can be written by plugging one of the points into the formula with the calculated slope to find the y-intercept (b). For the second line through points (-2, 2) and (1, 7), the slope is (7 - 2)/(1 - (-2)) = 5/3. An equation in point-slope form (y - y1 = m(x - x1)) can be written using one of the points and the calculated slope. Lines are parallel if their slopes are the same, and they are perpendicular if the product of their slopes is -1. If the slopes are neither identical nor negative reciprocals, the lines are neither parallel nor perpendicular.