Final answer:
To find the equation of a line that is parallel to y = 2x + 6 and passes through (-2, -3), the equation is y = 2x + 1. To find the equation of a line that is perpendicular to y = -3x - 3 and passes through (-2, -3), the equation is y = (1/3)x - 5/3. The lines y = -x - 5 and 5x - 5y = 25 are not perpendicular. The equation in point-slope form for the line through (-6, 9) with slope 3 is y - 9 = 3(x + 6).
Step-by-step explanation:
To find the equation of a line in slope-intercept form that is parallel to the graph of y = 2x + 6 and passes through the point (-2, -3), we can use the fact that parallel lines have the same slope. The slope of the given line is 2, so the slope of the parallel line is also 2. Plugging the slope (2) and the coordinates of the point (-2, -3) into the slope-intercept form equation y = mx + b, we get y = 2x + 1.
To find the equation of a line in slope-intercept form that is perpendicular to the graph of y = -3x - 3 and passes through the point (-2, -3), we can use the fact that perpendicular lines have slopes that are negative reciprocals. The slope of the given line is -3, so the slope of the perpendicular line is 1/3. Plugging the slope (1/3) and the coordinates of the point (-2, -3) into the slope-intercept form equation y = mx + b, we get y = (1/3)x - 5/3.
To determine if the lines y = -x - 5 and 5x - 5y = 25 are perpendicular, we can check if their slopes are negative reciprocals. The slope of the first line is -1 and the slope of the second line can be found by rearranging the equation to y = 5x - 25 and comparing the coefficient of x, which is 5. The slopes are not negative reciprocals, so the lines are not perpendicular.
To write an equation in point-slope form of the line through point (-6, 9) with slope 3, we can use the point-slope form equation y - y1 = m(x - x1). Plugging the values (-6, 9) for (x1, y1) and 3 for m, the equation becomes y - 9 = 3(x + 6).