Write a linear equation in slope intercept form for each of the following:

A line perpendicular to y=x+2 that passes through the origin.

A line perpendicular to y = -1/2x + 5 that passes through the points (4,1) and (1,-5).

A line that is the perpendicular bisector of a segment whose endpoints are (3,5) and (7,1).

determine the slope of the line you want

use point-slope form
convert to slope-intercept

#2 is thus
slope of given line is -1/2
perp. line has slope 2
y-1 = 2(x-4)
y = 2x-7
Luckily, that line also passes through (1,-5)

To write a linear equation in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept, we need to follow a few steps.

1. Find the slope of the given line.
2. Use the negative reciprocal of the slope to find the slope of the line perpendicular to the given line.
3. Use the point-slope formula or the two-point formula to find the equation of the line.

Now let's apply these steps to find the equations for each of the given lines.

1. A line perpendicular to y = x + 2 that passes through the origin.
Step 1: The given line has a slope of 1 (since it is in the form y = mx + b, where m = 1). To find a line perpendicular to this, we need the negative reciprocal, which is -1/1 or simply -1.
Step 2: We know the line passes through the origin (0, 0), which means that the y-intercept is 0 (b = 0).
Step 3: Using the slope-intercept form, the equation of the line is y = -x + 0, or simply y = -x.

2. A line perpendicular to y = -1/2x + 5 that passes through the points (4,1) and (1,-5).
Step 1: The given line has a slope of -1/2.
Step 2: We can choose any of the given points (4,1) or (1,-5) to find the y-intercept. Let's use the point (4,1).
Using the point-slope formula: (y - y1) = m(x - x1), where (x1, y1) is (4,1) and m is the negative reciprocal of the slope (-1/(-1/2)), which simplifies to 2.
Substituting the values in the point-slope form, we get: (y - 1) = 2(x - 4).
Simplifying further, we get: y - 1 = 2x - 8.
Rearranging the equation in slope-intercept form, we get: y = 2x - 7. This is the equation of the line required.

3. A line that is the perpendicular bisector of a segment whose endpoints are (3,5) and (7,1).
Step 1: To find the slope of the given segment, we use the formula: slope (m) = (y2 - y1) / (x2 - x1), where (x1, y1) = (3,5) and (x2, y2) = (7,1).
Substituting the values, we get: slope = (1 - 5) / (7 - 3) = -4 / 4 = -1.
Step 2: The line perpendicular to the given segment will have a slope of the negative reciprocal to -1, which is 1.
Step 3: To find the midpoint of the given segment, we use the formula: midpoint = ( (x1 + x2) / 2, (y1 + y2) / 2), where (x1, y1) = (3,5) and (x2, y2) = (7,1).
Substituting the values, we get: midpoint = ( (3 + 7) / 2, (5 + 1) / 2) = (5, 3).
Now we have the midpoint (5, 3) and the slope (1). Using the point-slope formula, we can write the equation of the line:
(y - y1) = m(x - x1), where (x1, y1) is (5, 3) and m is 1.
Substituting the values, we get: (y - 3) = 1(x - 5).
Simplifying further, we get: y - 3 = x - 5.
Rearranging the equation in slope-intercept form, we get: y = x - 2. This is the equation of the line required.

I hope this helps! Let me know if you have any further questions.