Question

Passing through the point (4,-1) and perpendicular to the line y = -6x - 4.

Answer 1

Our answer is y = 1/6x - 5/3.

`\Large\texttt{Explanation}`

We are asked to find the equation of the line that:

• Passes through (4,-1)
• Is perpendicular to the line y = -6x - 4

K E Y I N F O

• Perpendicular lines have slopes that are opposite reciprocals of each other.

The slope of y = -6x - 4 is -6; the opposite reciprocal of that is 1/6.

So, the line that's perpendicular to y = -6x - 4 will have a slope of 1/6.

FINDING THE EQUATION

To find the line's equation, use the point-slope formula -

• 
  `\bf{y-y_1=m(x-x_1)}`

Where -

• m = slope
• 
  `\bold{(x_1,y_1)}` is a point on the line

PUT IN THE VALUES

`\begin{gathered}\sf{y-y_1=m(x-x_1)}\\\sf{y-(-1)=\cfrac{1}{6}(x-4)}\\\sf{y+1=\cfrac{1}{6}(x-4)}\\\sf{y+1=\cfrac{1}{6}x-\cfrac{1}{6}*\cfrac{4}{1}}\\\sf{y+1=\cfrac{1}{6}x-\cfrac{4}{6}}\\\sf{y=\cfrac{1}{6}x-\cfrac{2}{3}-1}\\\sf{y=\cfrac{1}{6}x-\cfrac{2}{3}-\cfrac{3}{3}}\\\boxed{\sf{y=\cfrac{1}{6}x-\cfrac{5}{3}}}\end{gathered}`

`\therefore` y = 1/6x - 5/3

Answer 2

y =
`(1)/(6)` x -
`(5)/(3)`

Explanation:

the equation of a line in slope- intercept form is

y = mx + c ( m is the slope and c the y- intercept )

given the equation of the line

y = - 6x - 4 ← in slope- intercept form

with slope m = - 6

given a line with slope m then the slope of a line perpendicular to it is

`m_(perpendicular)` = -
`(1)/(m)` = -
`(1)/(-6)` =
`(1)/(6)` , then

y =
`(1)/(6)` x + c ← is the partial equation

to find c, substitute (4, - 1 ) for x and y into the partial equation

- 1 =
`(1)/(6)` (4) + c =
`(4)/(6)` + c =
`(2)/(3)` + c ( subtract
`(2)/(3)` from both sides )

-
`(5)/(3)` = c

y =
`(1)/(6)` x -
`(5)/(3)` ← equation of line