Write an equation in point-slope form of the line that passes through the given points:

1. (5,-6), (1,-7)
2. (6,-3), (-1,9)

3. write an equation of a line that is perpendicular to y=-3x+5 and passes through (4,3).

for #1 and #2

first find the slope, that gives you the m.
Now pick one of the given points and sub those values into
y = mx + b
Solve for b and rewrite the equation.

I am sure that your teacher and/or your textbook has given you several examples of this type of fundamental question.

#3.
The line perpedicular to the given line must have slope of 1/3

so new equation:
y = (1/3)x + b
sub in (4,3)
3 = (1/3)(4) + b
I usually get rid of my fractions by multiplying by whatever the denominator is
9 = 4 + 3b
b = 5/3

final equation : y = (1/3)x + 5/3

To write an equation in point-slope form of a line that passes through two given points, we need to use the formula:

y - y₁ = m(x - x₁)

where (x₁, y₁) are the coordinates of one point on the line, m is the slope of the line, and (x, y) are the coordinates of any other point on the line.

1. (5, -6) and (1, -7):
First, we need to find the slope (m) using the formula:

m = (y₂ - y₁) / (x₂ - x₁)

m = (-7 - (-6)) / (1 - 5)

m = (-7 + 6) / (-4)

m = -1 / (-4)

m = 1/4

Now, substituting the coordinates of the point (5, -6) into the equation:

y - (-6) = 1/4(x - 5)

y + 6 = 1/4(x - 5)

This is the equation in point-slope form for the line passing through the points (5, -6) and (1, -7).

2. (6, -3) and (-1, 9):
Calculate the slope (m):

m = (9 - (-3)) / (-1 - 6)

m = 12 / (-7)

m = -12 / 7

Using the point-slope form equation and the point (6, -3):

y - (-3) = -12/7(x - 6)

y + 3 = -12/7(x - 6)

This is the equation in point-slope form for the line passing through the points (6, -3) and (-1, 9).

3. To find an equation of a line perpendicular to y = -3x + 5 and passing through (4, 3), we need to determine its slope. The slope of a line perpendicular to another line is the negative reciprocal of the original line's slope.

The given equation is y = -3x + 5, so its slope is -3. The negative reciprocal of -3 is 1/3.

Using the point-slope form equation and the point (4, 3):

y - 3 = 1/3(x - 4)

y - 3 = 1/3x - 4/3

Multiply through by 3 to eliminate the fraction:

3(y - 3) = x - 4

3y - 9 = x - 4

Rearrange the equation to get the slope-intercept form (y = mx + b):

3y = x - 4 + 9

3y = x + 5

Divide through by 3:

y = (1/3)x + 5/3

This is the equation in slope-intercept form for the line perpendicular to y = -3x + 5 and passing through the point (4, 3).