a) To write the equation of the line that passes through the points (-2,-6) and (1,0) using point-slope form, we need to first find the slope of the line. The slope can be calculated as:
slope = (y2 - y1) / (x2 - x1)
slope = (0 - (-6)) / (1 - (-2))
slope = 2
Now that we know the slope, we can use the point-slope formula to write the equation of the line:
y - y1 = m(x - x1)
y - 0 = 2(x - 1)
Simplifying this equation, we get:
y = 2x - 2
Therefore, the equation of the line that passes through the points (-2,-6) and (1,0) is y = 2x - 2.
b) To write the equation of the line that passes through the point (-3,1) and is parallel to the line y = 4x + 3, we know that the slope of the new line will be the same as the slope of the given line, which is 4. So, we can use the point-slope formula again:
y - y1 = m(x - x1)
y - 1 = 4(x - (-3))
Simplifying this equation, we get:
y = 4x + 13
Therefore, the equation of the line that passes through the point (-3,1) and is parallel to the line y = 4x + 3 is y = 4x + 13.
c) To write the equation of the line that passes through the point (0, -6) and is perpendicular to the line -5x + y = 4, we need to first find the slope of the line -5x + y = 4. We can rewrite this equation in slope-intercept form to find its slope:
y = 5x + 4
The slope of this line is 5. Since the line we want to find is perpendicular to this line, its slope will be the negative reciprocal of 5, which is -1/5. Now we can use the point-slope formula to write the equation of the line:
y - y1 = m(x - x1)
y - (-6) = (-1/5)(x - 0)
Simplifying this equation, we get:
y = (-1/5)x - 6
Therefore, the equation of the line that passes through the point (0, -6) and is perpendicular to the line -5x + y = 4 is y = (-1/5)x - 6.