Final answer:
The equations of the lines parallel and perpendicular to the given line -5x+9y=4 passing through the point (-6,1) are y - 1 = (5/9)(x + 6) for the parallel line and y - 1 = (-9/5)(x + 6) for the perpendicular line.
Step-by-step explanation:
The given equation -5x+9y=4 represents a line with a slope and y-intercept determined by the coefficients of x and y. To find a parallel line equation that passes through the point (-6,1), we need a line with the same slope as the original line. Rearranging the original equation in slope-intercept form (y = mx + b), we get 9y = 5x + 4 and then y = (5/9)x + (4/9). So, the slope (m) is 5/9.
For the parallel line through (-6,1), we use the point-slope form: y - y1 = m(x - x1), giving us y - 1 = (5/9)(x + 6).
To find the perpendicular line equation that passes through (-6,1), we need the negative reciprocal of the original slope, which is -9/5. Using the point-slope form again, we have: y - 1 = (-9/5)(x + 6).