To write the equation of a line we need either (a) 2 points on the line or (b) the slope of the line and one point that lies on the line.
In this case, we will use the one given point: (12,-6), and the slope calculated to be perpendicular to the given line: y=4x+1.
The equation given is in slope-intercept form: y = mx + b, so we can immediately deduce that its slope is m=4. Slope is defined as the rise over the run, or change in y over change in x, of the line. A perpendicular line to the one given will then have the negative reciprocal as its slope, so that it is decreasing where the other is increasing, and has a rise of -1/4 for every rise of 4 in the other. This means the new line's slope will be -1/4.
Now we can write the equation of the new line using the point-slope formula: (y-y1)=m(x-x1), where m is our new slope and (x1,y1) is the given point.
(y-(-6))= -1/4(x-12)
Simplifying algebraically,
y+6 = -x/4 + 3
y = -x/4 -3
Which is now the slope-intercept form of the new line.
