Question

How do you write an equation of a line passing through (5, -3), perpendicular to `y=6x + 9`?

Answer 1

See a solution process below:

Explanation:

The equation in the problem is in slope-intercept form. 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.

`y = color(red)(6)x + color(blue)(9)`

The slope of this line is: `color(red)(m = 6)`

Let's call the slope of a perpendicular line: `m_p`

The formula for the slope of a perpendicular line is: `m_p = -1/m`

Substituting gives us:

`m_p = -1/6`

We can now use the point-slope formula to find an equation of the line from the problem. 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.

Substituting the slope we calculated and the values from the point in the problem gives:

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

`(y + color(red)(3)) = color(blue)(-1/6)(x - color(red)(5))`

If we want the equation in slope-intercept form we can solve for `y`:

`y + color(red)(3) = (color(blue)(-1/6) xx x) - (color(blue)(-1/6) xx color(red)(5))`

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

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

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

`y + 0 = -1/6x + 5/6 - (6/6 xx 3)`

`y = -1/6x + 5/6 - 18/6`

`y = color(red)(-1/6)x - color(blue)(13/6)`