Question

What is the equation of a line that is perpendicular to - x + 2y = 4 and passes through the point (-2,1)?

A. y = -2x - 3

B. y = 2x + 4

C. y = x + 2

D. y = 2x - 4

Answer 1

A. `y=-2x-3`.

Explanation:

First, rewrite the equation in slope-intercept form: `y=mx+b`.

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

So the slope of the given line is `m=1/2`.

Lines that are perpendicular have slopes that are negative reciprocals of each other. Meaning, if a line has slope `m`, then a line perpendicular to this has slope `m^"*"=(-1)/m`. That means the slope of our perpendicular line is

`m^"*"=(-1)/(1//2)=-2`.

Knowing the new slope and a point, we can find an equation for our perpendicular line by using the slope-point equation `y-y_1=m(x-x_1)`, or by plugging in the given `(x,y)` point (and the new slope `m^"*"`) into `y=mx+b` to find `b` for the new line.

`y-y_1=m(x-x_1)`
`=>y-1=-2[x-(-2)]`
`=>y-1=-2[x+2]`
`=>y=-2x-3`

or

`y=mx+b`
`=>1=-2(-2)+b`
`=>1=4+b`
`=>b=-3`

`:.y=mx+b` becomes `y=-2x-3`.