Question

How do you determine the equation of a line passing through (2, -3) that is perpendicular to the `4x-y=22`?

Answer 1

`y + 3 = -1/4x + 1/2`

or

`y = -1/4x - 5/2`

Explanation:

To determine this perpendicular line we can use the point-slope formula. We already have a point (2, -3), now we need to determine the slope. The slope of a perpendicular line is the negative inverse of the line it is perpendicular to. If we convert the equation we are given to the slope-intercept form we will have a slope we can take the negative inverse of.

`4x - y = 22`

`4x - y color(red)( + y - 22) = 22color(red)( + y - 22)`

`4x - 22 = y`

`y = 4x - 22`

The slope-intercept form is `color(red)(y = mx + b)` where `color(red)(m)` is the slope. Therefore the slope of the line we were give is `m = 4`

Given a slope `color(red)(m)` the negative inverse is `color(red)(-1/m)`

For our slope of `m = 4` the negative inverse is `-1/4`

Now we can use this slope and the point we were given and use the point-slope formula to find the equation of the perpendicular line.

The point-slope formula states: `color(red)((y - y_1) = m(x - x_1))`
Where `color(red)(m)` is the slope and `color(red)((x_1, y_1))` is a point the line passes through.

Substituting the information we have gives:

`y - -3 = -1/4(x - 2)`

`y + 3 = -1/4x + 2/4`

`y + 3 = -1/4x + 1/2`

If we want to convert to the more standard slope-intercept form we would get:

`y + 3 color(red)( - 3) = -1/4x + 1/2 color(red)( - 3)`

`y = -1/4x + 1/2 - 6/2`

`y = -1/4x - 5/2`