Can Someone check my math?

Question

Determine the equation of the line from each of the following line from each of the following clues.Express your answer in the form Ax+By+C=0.
a)The line through the point A(4,2) and B(6,-2)
b)The line parallel to y=9x+4 with y-intercept -7
c)The line perpendicular to y= -1/3x-5 and that passes through the point D(2,5).
d)The line parallel to 3x+y=0, that has the same x-intercept as y=3x-9
e)The line with the same y- intercept as 3x-4y+8=0 and the same x-intercept as 2x +5y-8=0
f)The line perpendicular to x+2y-3=0, but with the same y-intercept
g)The line parallel to y-6=0 but with y-intercept 10

Answer 1

a)The line through the point A(4,2) and B(6,-2)

Given  points A(4,2) ; B(6, -2)

Let A(4,2)= (x1,y1); B(6, -2) = (x2 , y2)

Line equation is (y-y1) = M (x-x1)---------(1)

M = slope

M = y2-y1 / x2-x1

M = - 2 - 4 / 6 - 4

M = -6 / 2

M = -3

substitute M = - 3 in the equation(1)

                         ( y- 2 ) = - 3 ( x- 4)

                         ( y - 2 ) = -3x +12

                          y - 2 = -3x+12

Add 3x to each side

 y-2+3x = -3x+12+3x

3x+y-2 = 12

Subtract 12 from each side

3x+y-2-12 = 12-12

3x+y -14 = 0

Line equation = 3x+y-14

 

Answer 2

b) Slope of the line y = 9x + 4 is 9.

If two vertical lines are parallel then the slope of one is equal to the slope of other.

Therefore slope of a parallel line is also 9.

Use intercept form of line equation is y = mx + c.

Substitute parallel line slope m = 9 and  y-intercept c = -7.

Therefore y = 9x - 7 is the required parallel line.

c) Slope of the line y = -1/3x - 5 is -1/3.

If two vertical lines are perpendicular then the slope of one is negative reciprocal of the slope of other.

Slope of the perpendicular line is -1/m = -1/(-1/3) = 3.

Use general point slope form of line equation since point is given.

(y - y₁ ) = m (x - x₁ ) where (x₁, y₁ ) = (2,5) and m = 3.

(y - 5) = 3(x - 2)

y - 5 = 3x - 6

y = 3x - 6 + 5

y = 3x - 1 is the required perpendicular line.