Question:
Find the equation of the line that is perpendicular to y = x + 7 and goes through the point (-2,3).
Solution.
The slope-intercept form for a line is given by the following formula:
y = mx+b
where m is the slope of a line and b is the y-coordinate of the y-intercept. Notice that if y= x+7 then the slope m of this line is m= 1. Now, perpendicular lines have slopes that are the opposite of the reciprocal of each other. In this case, the slope of the first line is m = 1, then the reciprocal of m = 1 is 1/1 so the opposite of the reciprocal is therefore -1. Thus, we can conclude that the slope of the line that is perpendicular to the given line is m* = -1. That is, this line has an equation of the form:
y = -1x + b
now, to find b, take any point of the given perpendicular line, for example, take the point (x,y) = (-2,3), replace it in the above equation and solve for b:
3 = -1(-2) + b
then
b = 3 -2 = 1
then, we can conclude that the equation of the perpendicular line is
y = -x +1