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Use the point slope to write an equation of:

a)The line that passes through the points (-2,-6) and (1,0)
b) The line that passes through the point (-3,1) and is parallel to the line y = 4x + 3.
C) The line that passes through the point (0, -6) and is perpendicular to the line -5x + y = 4.

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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.
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