Write an equation of a line perpendicular to y= 2x+3 through (-2,4)

Answer 1

Perpendicular lines have opposite, reciprocal slopes.

y = mx + b

4 = (-1/2)(-2) + b

4 = 1 + b

b = 3

y = (-1/2)x + 3

Answer 2

The "slope intercept form" of a linear equation is y = mx + b.

m = gradient = change in y / change in x and b = y intercept = point (0, b).

Your line is y = 2x + 3 which is already in this form (if not, rearrange). In this case, m = 2, b = 3.

For two perpendicular lines, the key property is that m1.m2 = -1.

As m1 = 2, we have 2.m2 = -1 => m2 = -¹/₂.

Hence, the general equation for the infinite number of lines perpendicular to y = 2x + 3 is y = -¹/₂ x + n

But now we need the particular line that goes through (-2, 4).

In this case, y = -¹/₂ x + c and we need to figure out c. We can do so by substituting the point values to get:

(4) = -¹/₂ (-2) + c

=> 4 = 1 + c

=> c = 3

Hence required perpendicular line has the equation y = -¹/₂ x + 3 which is equivalent to 2y = 6 - x

Don't forget to double-check your solution that it actually works (it does). I recommend Desmos graphing calculator if you want to see what this looks like.