Math CentralQuandaries & Queries


Question from Jesse, a student:

How do you write an equation of a line that is perpendicular to y+2x+3 and passes through (3,4);
I don't understand the steps of how to do this????

Hi Jesse.

I presume you meant to write y = 2x + 3.

If the equation of a line is written in the form

y = mx + b

then m is the slope so the first line has a slope of 2.

Parallel lines have the same slope and perpendicular lines have negative reciprocal slopes. That is, if
y = mx + b and y = kx + c are perpendicular then k = -1/m. Thus a line which is perpendicular to your line has a slope of -1/m = -1/2.

Hence a line perpendicular to your first line has an equation of the form

y = -1/2 x + b for some number b.

Since you know that this line passes through the point (3,4), you know that when x = 3 then y = 4. So you can put these values into the last equation to find the value of b.

4 = -1/2 (3) + b

Solve for b and you have all the pieces you need to write the equation of the perpendicular line.

Stephen and Penny

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