Question

How do you write the standard form of the equation of the line that is perpendicular to y=-4/3x-7 and contains (-5,-7)?

Answer 1

The standard form equation is `3x - 4y = 13`.

Explanation:

The slope of our line is `color(red)(-\frac{4}{3})`. The slope of any perpendicular line is given by the negative reciprocal.

`m' = -\frac{1}{m}`

`m' = \frac{1}{color(red)(-\frac{4}{3})}`

`m' = \frac{3}{4}`

The second condition is that the line contains `(color(blue)(-5), color(green)(-7))`.

`y = 3/4x + b`

Substitute `color(blue)(x = -5)`, `color(green)(y = -7)`.

`color(green)(-7) = 3/4(color(blue)(-5)) + b`

`color(green)(-7) = -15/4 + b`

`-13/4 = b`

So in slope-intercept form, the equation of our line is

`y = 3/4x - 13/4`

but the question requires it in standard form. This means that it should be in the form

`Ax + By = C`

where `A`, `B`, and `C` are all integers and `A` is positive. In our equation, we will multiply by the lowest common denominator to eliminate the denominator.

`4y = 3x - 13`

`<=> 3x - 13 = 4y`

`<=> 3x - 4y = 13`