Michael D. answered • 06/14/19
U of M Math Teaching Program; Common Core Concepts, MS Purdue, SAT
The general equation we use for a line is f(x) = mx +b.
The slope of our line is the ratio of the change in the function relative to a change in the argument x and is the constant m in our expression. The intercept is by definition where the line crosses the f(0) axis.
lets look at the line y=x the slope is positive 1....what lines are perpendicular to this line....
any line with slope -1
Now lets look at the line y=(1/2) x we can see that the lines that are perpendicular to this line have a rise over run of negative 2...a negative of the reciprocal of the original slope.
so in general given the slope intercept form of a line, all lines with slope minus the reciprocal the the original slope are perpendicular to our original line.
our original line is expressed x +9y =7 or y= -1/9 x +7/9...
The slope of our required lines that are perpendicular to this line is then 9....
our new lines are the set y=9x +b
now we wish to match b to the y intercept of the line 2x-2y=4.
Writing this in slope intercept form we have 2y= 2x-4 or y= x-2 and b for our perpendicular line is -2, the intercept.
Our line is then y= 9x -2 or 2= 9x-y
One can also write the general perpendicular lines as 9x-y = c and compare this to our general lines
x +9y =b
note the second set of lines transforms into the first set of lines on mapping x to y and y to( -x) ....This is a rotation through 90 degrees of our original set of lines with coordinates (x,y) mapping to (y, -x)
