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

1 Answer
Dec 18, 2016

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