20. Write the equation of a line that is perpendicular to the given line and that passes through the given point

y-3=8/3(x+2); (-2,3)

(is there a formula to follow?)

yes, the formula they appear to follow is this:

given: the slope m, and a point (a,b)
equation of line: y-b = m(x-a)

so the slope of the given line is 8/3
perpendicular lines have slopes that are negative reciprocals of each other.
So the slope of the new perpendicular line must be -3/8

follow my formula to state the equation.

Yes, there is a formula you can follow to find the equation of a line that is perpendicular to a given line and passes through a given point.

To do this, you need to find the slope of the given line and then use the negative reciprocal of that slope to determine the slope of the line perpendicular to it. Once you have the slope, you can use the point-slope form of a linear equation to find the equation of the perpendicular line.

Let's go through the steps:

1. Begin with the given equation of the line: y - 3 = (8/3)(x + 2)
Rewrite it in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept:
y = (8/3)x + 8/3 + 3
y = (8/3)x + 17/3

2. Find the slope of the given line. The slope of a line in slope-intercept form (y = mx + b) is the coefficient of x, which in this case is 8/3. Therefore, the slope of the given line is 8/3.

3. Determine the negative reciprocal of the slope. To find the slope of a line perpendicular to the given line, take the negative reciprocal of the slope. The negative reciprocal of 8/3 is -3/8.

4. Use the point-slope form of a linear equation to find the equation of the perpendicular line. The point-slope form of a linear equation is given by:
y - y1 = m(x - x1)
where (x1, y1) is the given point and m is the slope of the line.

Substituting the point (-2, 3) and the slope -3/8 into the point-slope form, we have:
y - 3 = (-3/8)(x - (-2))
y - 3 = (-3/8)(x + 2)
y - 3 = (-3/8)x - 3/4

5. Simplify the equation by multiplying through by 8 to eliminate fractions:
8(y - 3) = -3(x) - (3/4)(8)
8y - 24 = -3x - 6
8y = -3x + 18

Therefore, the equation of the line that is perpendicular to the given line and passes through the point (-2, 3) is 8y = -3x + 18.