Find (a) the equation of a line that is parallel to the given line and includes the given point, and (b) the equation of a line that is perpendicular to the given line through the given point. Write both answers in slope-intercept form.

y = -4x + 2, (5, 4)

(a) You want a line with slope -4 that passes through (5,4)

The equation will have the form
y = -4x + b
Now substityute x = 5 and y = 4 and solve for b.
4 = -20 + b
b = 24
y = -4x + 24

(b) You want a line with slope 1/4 that passes through (5,4)
Follow the same procedure:
y = x/4 + b'
Let x = 5, y = 4 and solve for b'
4 = 5/4 + b'
b' = 11/4
y = (x/4) + 11/4

To find the equation of a line that is parallel to the given line and includes the given point, you can use the fact that parallel lines have the same slope.

Step 1: Determine the slope of the given line. The given line is in slope-intercept form, y = mx + b, where m represents the slope. In this case, the slope is -4.

Step 2: Use the slope and the given point to find the equation of the parallel line using the point-slope form, y - y1 = m(x - x1), where (x1, y1) represents the given point.

Using the point (5, 4):

y - 4 = -4(x - 5)

Simplifying the equation:

y - 4 = -4x + 20

y = -4x + 24

So, the equation of the line that is parallel to y = -4x + 2 and includes the point (5, 4) is y = -4x + 24.

To find the equation of a line that is perpendicular to the given line through the given point, you can use the fact that perpendicular lines have slopes that are negative reciprocals of each other.

Step 1: Determine the slope of the given line. In this case, the slope is -4.

Step 2: Find the negative reciprocal of the slope. In this case, the negative reciprocal of -4 is 1/4.

Step 3: Use the negative reciprocal slope and the given point to find the equation of the perpendicular line using the point-slope form.

Using the point (5, 4):

y - 4 = (1/4)(x - 5)

Simplifying the equation:

y - 4 = (1/4)x - 5/4

y = (1/4)x + 11/4

So, the equation of the line that is perpendicular to y = -4x + 2 and passes through the point (5, 4) is y = (1/4)x + 11/4.