What is the slope of the line that is perpendicular to the line whose equation is 2x y 4?

The product of the slope of a line with the slope of a line perpendicular to it is -1.

Or to put it another way, if a line has an equation

y = mx + c

a line perpendicular to it will have a gradient m' such that

mm' = -1

⇒ m' = -1/m

and an equation of

y = -1/m x + b

which can be rearranged to give

my + x = d

(where d = mb).

There are symbols missing from the question, which I suspect are a plus (+) and an equals (=). They can be inserted in two different ways, leading to two different solutions:

• 2x + y = 4

⇒ y = -2x + 4

⇒ gradient of perpendicular line is -1/-2 = 1/2

⇒ perpendicular line is:

y = 1/2 x + b

⇒ 2y = x + c

• 2x = y + 4

⇒ y = 2x - 4

⇒ gradient of perpendicular line is -1/2 = -1/2

⇒ perpendicular line is:

y = -1/2 x + b

⇒ 2y + x = c

Note: there is no one perpendicular line, the value of the constant c will vary depending upon what point the line must pass through.