Question

Write an equation for a line perpendicular to `y = 1/4x + 6` and going through the point (-2, 9)#?

Answer 1

See a solution process below:

Explanation:

The equation for the line 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.

Therefore the slope of this line is: `color(red)(1/4)`

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

The formula for find the slope of a perpendicular line is:

`m_p = -1/m`

Substituting the slope we found above gives:

`m_p = -1/(1/4) = -4`

We can now use the point-slope formula to write an equation for the line. 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)(9)) = color(blue)(-4)(x - color(red)(-2))`

`(y - color(red)(9)) = color(blue)(-4)(x + color(red)(2))`

We can also solve this equation for `y` to transform the equation into the slope-internet form:

`y - color(red)(9) = (color(blue)(-4) xx x) + (color(blue)(-4) xx color(red)(2))`

`y - color(red)(9) = -4x - 8`

`y - color(red)(9) + 9 = -4x - 8 + 9`

`y - 0 = -4x + 1`

`y = color(red)(-4)x + color(blue)(1)`