Write an equation for a line perpendicular

Question

Write an equation for a line perpendicular to 2 y + 4 x = 2 and passing through the point (2,-4)

Answer 1

The first step I would do toward solving this problem is to get the equation in slope-intercept form. This means isolating y as a function of x. Firstly you would subtract 4x from both sides, and be left with:

2y = 2 - 4x

In slope intercept form, the term with x always precedes the intercept, so you must "switch" the terms of 2 and -4x ; that leaves us with:

2y = -4x + 2

The next step involves dividing every term by 2 to render y alone:

y = -2x + 1

This is the equation of the line to which we want to find a perpendicular line. Remember that the slope of a line perpendicular to another line is the negative reciprocal of its slope. In slope intercept form, with an isolated y, the slope is the coefficient of the x term. So the negative reciprocal of the slope is the negative reciprocal of -2, which would be 1/2

So now that we know that the slope of the perpendicular line is 1/2, we must find the y-intercept.

I usually think of this in two ways. Firstly, you can sketch a graph and plot a point at (2,-4), and since you know the slope (rise/run) is 1/2, you can work backwards to find the y-intercept.

The algebraic way to do this is to put the point and slope into point-slope form: y-y₁ = m(x-x₁)

where m is the slope and (x₁,y₁) is your point. Plugging this in you get:

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

or

y+4 = 1/2(x-2)

Now you only have to solve for y in slope-intercept form

y+4 = (1/2)x - 1

y = (1/2)x - 5

This is your final answer, the equation of a line perpendicular to 2y + 4x = 2 and passing through the point (2,-4) in slope-intercept form is

y = (1/2)x - 5

Answer 2

2y + 4x = 2

2y = -4x + 2

y = -2x + 1

Use point-slope form

(y + 4) = (-1/2)(x - 2)