Question

What is the equation of the line perpendicular to `y =3x- 7` that contains (6, 8)?

Answer 1

`(y - 8) = -1/3(x - 6)`

or

`y = -1/3x + 10`

Explanation:

Because the line given in the problem is in the slope intercept form we know the slope of this line is `color(red)(3)`

The slope-intercept form of a linear equation is:

`y = color(red)(m)x + color(blue)(b)`

Where `color(red)(m)` is the slope and `color(blue)(b` is the y-intercept value.
This is a weighted average problem.

Two perpendicular lines have a negative inverse slope of each other.

The line perpendicular to a line with slope `color(red)(m)` has a slope of `color(red)(-1/m)`.

Therefore, the line we are looking for has a slope of `color(red)(-1/3)`.

We can now use the point-slope formula to find the equation of the line we are looking for.

The point-slope formula states: `(y - color(red)(y_1)) = color(blue)(m)(x - color(red)(x_1))`

Where `color(blue)(m)` is the slope and `color(red)(((x_1, y_1)))` is a point the line passes through.

We can substitute the slope we calculate and the point we were given to give the equation we are looking for:

`(y - color(red)(8)) = color(blue)(-1/3)(x - color(red)(6))`

If we want to put this in slope-intercept form we can solve for `y`:

`y - color(red)(8) = color(blue)(-1/3)x - (color(blue)(-1/3) xx color(red)(6)))`

`y - color(red)(8) = color(blue)(-1/3)x - (-2)`

`y - color(red)(8) = color(blue)(-1/3)x + 2`

`y - color(red)(8) + 8 = color(blue)(-1/3)x + 2 + 8`

`y - 0 = color(blue)(-1/3)x + 10`

`y = -1/3x + 10`